Generating functions of Littlewood-Richardson coefficients

Yoshio Agaoka

Department of Mathematics, Faculty of Integrated Arts and Sciences Hiroshima University, Higashi-Hiroshima 739-8521, Japan e-mail address: agaoka@mis.hiroshima-u.ac.jp

Abstract

We give generating functions of the Littlewood-Richardson coefficients expressing the product of two Schur functions for the cases {λ_{1}, ⋯, λ_{m}}{μ_{1}} (m ≥ 1) and {λ_{1}, ⋯, λ_{m}}{μ_{1}, μ_{2}} (m ≤ 4). These formulas also give generating functions expressing the number of irreducible components of {λ}{μ} for these cases. As an application, we give a new decomposition formula of the product {λ_{1}, λ_{2}}{μ_{1}, μ_{2}}, expressed as a linear combination of some basic partitions whose coefficients move in some polytopes. Several conjectures on the generating function for the case {λ_{1}, ⋯, λ_{m}}{μ_{1}, μ_{2}} (m ≥ 5) and a simple formula expressing the Littlewood-Richardson polynomial for the case {λ} = {λ_{1}, λ_{2}}, {μ} = {μ_{1}, μ_{2}} are also stated.

1. Introduction.

The Littlewood-Richardson coefficient c_{λμ}^{ν} is the coefficient of S_{ν} in the product of two Schur function S_{λ} and S_{μ}, i.e., S_{λ}S_{μ} = ∑ c_{λμ}^{ν}S_{ν}. (Hereafter, we use the classical notation {λ}{μ} instead of S_{λ}S_{μ}). This coefficient is also equal to the multiplicity of V_{ν} in the tensor product V_{λ} ⊗ V_{μ}, where V_{λ} etc. means the irreducible representation space of the general linear group GL(V) corresponding to the partition {λ} etc. The coefficient c_{λμ}^{ν} also appears in many fields of mathematics such as combinatorics, representation theory, algebraic geometry, etc. The value c_{λμ}^{ν} can be calculated by the Littlewood-Richardson rule, which requires a combinatorial procedure on Young diagrams. Other several methods to calculate c_{λμ}^{ν} are also known. For example, it is known that the coefficient c_{λμ}^{ν} can be expressed as a number of lattice points in some convex polytopes (cf. [5], [6], [12]). But

2000 Mathematics Subject Classification. Primary 20G05; Secondary 05E05, 05A15, 52B20.

Key words and phrases. Littlewood-Richardson coefficient, Schur function, tensor product, generating function.

in general it is almost impossible to express the individual value c_{λμ}^{ν} as a function of {λ}, {μ} and {ν}, because it requires many case by case examinations. (Exceptionally, some special formulas are known. See [19], [22], [23], [24] etc.)

The purpose of this paper is to give several new formulas expressing the explicit values of c_{λμ}^{ν}. Our main tool is a “generating function”, which seems to be the most natural language to describe the values c_{λμ}^{ν}. One generating function contains all informations on c_{λμ}^{ν}’s when the depths of {λ} and {μ} are fixed. This phenomenon is surprising because the Littlewood-Richardson rule itself is a combinatorial algorithm on Young diagrams, and because this result means that the totality of such combinatorial calculations is condensed into one generating function.

There are already several attempts to calculate generating functions of c_{λμ}^{ν} such as [4], [17], etc. But it seems that the main interests of these papers are devoted to find an efficient algorithm to calculate them, not directed to their explicit forms nor their properties. But after some calculations, we are convinced that generating functions possess several peculiar properties deserve to more attention. The main purpose of this paper is to give their explicit expressions and to explain their properties, mainly on two types of products {λ_{1}, ⋯, λ_{m}} {μ_{1}} (m ≥ 1) and {λ_{1}, ⋯, λ_{m}}{μ_{1}, μ_{2}} (m ≤ 4) (Theorems 1 and 4). We also state several conjectures for the second type product for m ≥ 5.

In addition, as one application of such generating functions, we give a new proof of the result of Stembridge [26], characterizing the pair (λ, μ) such that the product {λ}{μ} is multiplicity-free in case the depths of {λ} and {μ} ≤ 2 (Theorem 8). For these {λ} and {μ} we also give a quite simple formula expressing the Littlewood-Richardson polynomial (Theorem 9).

2. The case {λ_{1}, ⋯, λ_{m}}{μ_{1}}.

In this section, we first consider the generating function for the product {λ}{μ} = {λ_{1}, ⋯, λ_{m}}{μ_{1}}. In this case each component of the product {λ}{μ} consists of partitions of the form

{ν} = {|λ| + μ_{1} − |ν|, ν_{1}, ⋯, ν_{m}},

where |λ| = λ_{1} + ⋯ + λ_{m} and |ν| = ν_{1} + ⋯ + ν_{m}. The decomposition can be calculated by Pieri’s formula, which is a special case of the Littlewood-Richardson rule [25]. We express the multiplicity of {ν} in {λ}{μ} as c_{λμ}^{ν} and consider the following generating function expressing the decomposition of {λ}{μ}:

F_{m,1} = ∑_{λ,μ,ν} c_{λμ}^{ν} x_{1}^{ν_{1}} ⋯ x_{m}^{ν_{m}}q_{1}^{λ_{1}} ⋯ q_{m}^{λ_{m}}r_{1}^{μ_{1}}.

Namely the coefficient of x_{1}^{ν_{1}} ⋯ x_{m}^{ν_{m}}q_{1}^{λ_{1}} ⋯ q_{m}^{λ_{m}}r_{1}^{μ_{1}} of this function is equal to the multiplicity of {ν} in {λ}{μ}. (In the case {μ} = {μ_{1}}, the product {λ}{μ} is multiplicityfree and this coefficient is actually 0 or 1.) For fixed m, this single function contains all informations on the decomposition of the product {λ}{μ} for {λ} with depth≤ m and {μ} with depth≤ 1. We consider the problem expressing this generating function F_{m,1} in a simple form. The answer is given in the following theorem.

Theorem 1. We have

F_{m,1} = \frac{1}{(1 − q_{1})(1 − x_{1}q_{1}q_{2})(1 − x_{1}x_{2}q_{1}q_{2}q_{3}) ⋯ (1 − x_{1} ⋯ x_{m−1}q_{1} ⋯ q_{m})}

× (1 − r_{1})(1 − x_{1}q_{1}r_{1})(1 − x_{1}x_{2}q_{1}q_{2}r_{1}) ⋯ (1 − x_{1} ⋯ x_{m}q_{1} ⋯ q_{m}r_{1}).

Note that if we put x_{1} = ⋯ = x_{m} = 1 into these functions, we obtain the generating functions expressing the number of irreducible components of {λ}{μ}.

Corollary 2. The number of irreducible components of {λ_{1}, ⋯, λ_{m}}{μ_{1}} is equal to the coefficient of q_{1}^{λ_{1}} ⋯ q_{m}^{λ_{m}}r_{1}^{μ_{1}} in the following function:

1

(1 − q_{1})(1 − q_{1}q_{2})(1 − q_{1}q_{2}q_{3}) ⋯ (1 − q_{1} ⋯ q_{m})

× (1 − r_{1})(1 − q_{1}r_{1})(1 − q_{1}q_{2}r_{1}) ⋯ (1 − q_{1} ⋯ q_{m}r_{1}).

Example. We consider the case m = 3. In this case the function F_{3,1} is expanded as:

F_{3,1} = 1 + q_{1} + r_{1} + q_{1}^{2} + x_{1}q_{1}q_{2} + (1 + x_{1})q_{1}r_{1} + r_{1}^{2} + q_{1}^{3} + x_{1}q_{1}^{2}q_{2} + x_{1}x_{2}q_{1}q_{2}q_{3}

+ (1 + x_{1})q_{1}^{2}r_{1} + (x_{1} + x_{1}x_{2})q_{1}q_{2}r_{1} + (1 + x_{1})q_{1}r_{1}^{2} + r_{1}^{3} + · · · · · ·
+ (x_{1}x_{2} + x_{1}x_{2}x_{3} + x_{1}^{2}x_{2} + x_{1}^{2}x_{2}x_{3})q_{1}^{2}q_{2}q_{3}r_{1}^{2} + · · · · · · .

From the coefficients of q_{1}r_{1} and q_{1}^{2}q_{2}q_{3}r_{1}^{2}, we obtain the following decompositions:

{1}{1} = {2} + {1^{2}},
{21^{2}}{2} = {41^{2}} + {31^{3}} + {321} + {2^{2}1^{2}}.

If we put x_{1} = x_{2} = x_{3} = 1 into F_{3,1}, we obtain

1 + q_{1} + r_{1} + q_{1}^{2} + q_{1}q_{2} + 2q_{1}r_{1} + r_{1}^{2} + q_{1}^{3} + q_{1}^{2}q_{2} + q_{1}q_{2}q_{3} + 2q_{1}^{2}r_{1} + 2q_{1}q_{2}r_{1} + 2q_{1}r_{1}^{2} + r_{1}^{3} + ⋯ + 8q_{1}^{3}q_{2}^{2}q_{3}r_{1}^{4} + · · · · · · .

Hence, for example, we know that the number of irreducible components of {321}{4} is 8.

We give another decomposition formula of {λ_{1}, ⋯, λ_{m}}{μ_{1}} as a corollary of Theorem 1, which is similar to the decomposition formulas of plethysms stated in [1], [2].

Corollary 3. We have the following decomposition formula:

{λ_{1}, λ_{2}, ⋯, λ_{m}}{μ_{1}} =

∑_{\{ \begin{matrix}a_{1} + b_{1} = λ_{1} − λ_{2} \\ a_{2} + b_{2} = λ_{2} − λ_{3} \\ ⋯ & ⋯ & ⋯ \\ a_{m−1} + b_{m−1} = λ_{m−1} − λ_{m} \\ a_{m} + b_{m} = λ_{m} \\ a_{0} + a_{1} + ⋯ + a_{m} = μ_{1} \\ a_{i}, b_{j} ≥ 0 \end{matrix}} \begin{matrix}[ (a_{0} + b_{1})\{1, 0, ⋯, 0\} + (a_{1} + b_{2})\{1, 1, 0, ⋯, 0\} + · · · · · · \\ + (a_{m−1} + b_{m})\{1, 1, ⋯, 1, 0\} + a_{m}\{1, 1, ⋯, 1\} ] . \end{matrix}

Example. We consider the case {421}{2}. In this case the set of non-negative integers satisfying the conditions

\{ \begin{matrix}a_{1} + b_{1} = 4 − 2 = 2, \\ a_{2} + b_{2} = 2 − 1 = 1, \\ a_{3} + b_{3} = 1, \\ a_{0} + a_{1} + a_{2} + a_{3} = 2 \end{matrix}

is given as follows:

\begin{matrix}a_{0} & a_{1} & a_{2} & a_{3} & b_{1} & b_{2} & b_{3} & \{ν\} \\ 2 & 0 & 0 & 0 & 2 & 1 & 1 & \{621\} \\ 1 & 0 & 0 & 1 & 2 & 1 & 0 & \{521^{2}\} \\ 1 & 0 & 1 & 0 & 2 & 0 & 1 & \{52^{2}\} \\ 0 & 0 & 1 & 1 & 2 & 0 & 0 & \{42^{2}1\} \\ 1 & 1 & 0 & 0 & 1 & 1 & 1 & \{531\} \\ 0 & 1 & 0 & 1 & 1 & 1 & 0 & \{431^{2}\} \\ 0 & 1 & 1 & 0 & 1 & 0 & 1 & \{432\} \\ 0 & 2 & 0 & 0 & 0 & 1 & 1 & \{4^{2}1\} \end{matrix}

And the right column gives the desired decomposition of {421}{2}.

Now we give a proof of Theorem 1 and Corollary 3.

Proof of Theorem 1. We fix {λ} = {λ_{1}, λ_{2}, ⋯, λ_{m}} and {μ} = {μ_{1}}. To each row of the Young diagram {λ} we add a_{0}, a_{1}, ⋯, a_{m} new boxes with a_{0} + ⋯ + a_{m} = μ_{1} respectively as follows (this figure corresponds to the case m = 3).

From the Littlewood-Richardson rule the integers a_{i} must satisfy the following conditions:

(2.1)  \{ \begin{matrix}a_{1} ≤ λ_{1} − λ_{2}, \\ a_{2} ≤ λ_{2} − λ_{3}, \\ ⋯ & ⋯ \\ a_{m−1} ≤ λ_{m−1} − λ_{m}, \\ a_{m} ≤ λ_{m}. \end{matrix}

λ_{1} λ_{2} λ_{3}  a_{0} a_{1} a_{2} a_{3}

Then we have

F_{m,1} = ∑_{λ,μ,a_{i}} x_{1}^{λ_{2}+a_{1}}x_{2}^{λ_{3}+a_{2}} ⋯ x_{m−1}^{λ_{m}+a_{m−1}}x_{m}^{a_{m}}q_{1}^{λ_{1}}q_{2}^{λ_{2}} ⋯ q_{m}^{λ_{m}}r_{1}^{μ_{1}},

where a_{i} move in the above range (2.1) and satisfy a_{0} + ⋯ + a_{m} = μ_{1}. We here define non-negative integers b_{i} by

\{ \begin{matrix}a_{1} + b_{1} = λ_{1} − λ_{2}, \\ a_{2} + b_{2} = λ_{2} − λ_{3}, \\ ⋯ & ⋯ \\ a_{m−1} + b_{m−1} = λ_{m−1} − λ_{m}, \\ a_{m} + b_{m} = λ_{m}. \end{matrix}

Then we have

\{ \begin{matrix}λ_{1} = a_{1} + ⋯ + a_{m} + b_{1} + ⋯ + b_{m}, \\ λ_{2} = a_{2} + ⋯ + a_{m} + b_{2} + ⋯ + b_{m}, \\ ⋯ & ⋯ \\ λ_{m−1} = a_{m−1} + a_{m} + b_{m−1} + b_{m}, \\ λ_{m} = a_{m} + b_{m}. \end{matrix}

Hence we have

F_{m,1} = ∑_{a_{i},b_{i}} x_{1}^{a_{1}+···+a_{m}+b_{2}+···+b_{m}}x_{2}^{a_{2}+···+a_{m}+b_{3}+···+b_{m}} ⋯ x_{m−1}^{a_{m−1}+a_{m}+b_{m}}x_{m}^{a_{m}}

× q_{1}^{a_{1}+···+a_{m}+b_{1}+···+b_{m}}q_{2}^{a_{2}+···+a_{m}+b_{2}+···+b_{m}} ⋯ q_{m}^{a_{m}+b_{m}}r_{1}^{a_{0}+···+a_{m}}

= ∑_{a_{i},b_{i}} q_{1}^{b_{1}}(x_{1}q_{1}q_{2})^{b_{2}}(x_{1}x_{2}q_{1}q_{2}q_{3})^{b_{3}} ⋯ (x_{1} ⋯ x_{m−2}q_{1} ⋯ q_{m−1})^{b_{m−1}}

× (x_{1} ⋯ x_{m−1}q_{1} ⋯ q_{m})^{b_{m}}r_{1}^{a_{0}}(x_{1}q_{1}r_{1})^{a_{1}}(x_{1}x_{2}q_{1}q_{2}r_{1})^{a_{2}} ⋯
× (x_{1} ⋯ x_{m−1}q_{1} ⋯ q_{m−1}r_{1})^{a_{m−1}}(x_{1} ⋯ x_{m}q_{1} ⋯ q_{m}r_{1})^{a_{m}}

= \frac{1}{(1 − q_{1})(1 − x_{1}q_{1}q_{2})(1 − x_{1}x_{2}q_{1}q_{2}q_{3}) ⋯ (1 − x_{1} ⋯ x_{m−1}q_{1} ⋯ q_{m})}

× (1 − r_{1})(1 − x_{1}q_{1}r_{1})(1 − x_{1}x_{2}q_{1}q_{2}r_{1}) ⋯ (1 − x_{1} ⋯ x_{m}q_{1} ⋯ q_{m}r_{1}).

q.e.d.

Proof of Corollary 3. We consider the equality in the above proof:

F_{m,1} = ∑ x_{1}^{a_{1}+···+a_{m}+b_{2}+···+b_{m}}x_{2}^{a_{2}+···+a_{m}+b_{3}+···+b_{m}} ⋯ x_{m−1}^{a_{m−1}+a_{m}+b_{m}}x_{m}^{a_{m}}

× q_{1}^{a_{1}+···+a_{m}+b_{1}+···+b_{m}}q_{2}^{a_{2}+···+a_{m}+b_{2}+···+b_{m}} ⋯ q_{m}^{a_{m}+b_{m}}r_{1}^{a_{0}+a_{1}···+a_{m}}.

This term corresponds to the partition

{∗, a_{1} + ⋯ + a_{m} + b_{2} + ⋯ + b_{m}, a_{2} + ⋯ + a_{m} + b_{3} + ⋯ + b_{m}, · · · · · · , a_{m−1} + a_{m} + b_{m}, a_{m}}

in the product {λ_{1}, …, λ_{m}}{μ_{1}}, where

\{ \begin{matrix}λ_{1} = a_{1} + ⋯ + a_{m} + b_{1} + ⋯ + b_{m}, \\ λ_{2} = a_{2} + ⋯ + a_{m} + b_{2} + ⋯ + b_{m}, \\ ⋯ & ⋯ & ⋯ \\ λ_{m} = a_{m} + b_{m}, \\ μ_{1} = a_{0} + a_{1} + ⋯ + a_{m}. \end{matrix}

It is easy to see that the value of ∗ in the above term is equal to a_{0} + a_{1} + ⋯ + a_{m} + b_{1} + ⋯ + b_{m},and hence Corollary 3 follows immediately from these expressions. q.e.d.

3. The case {λ_{1}, ⋯, λ_{m}}{μ_{1}, μ_{2}}.

Next, in this section we consider the case {λ}{μ} = {λ_{1}, ⋯, λ_{m}}{μ_{1}, μ_{2}}. In this case the tensor product {λ}{μ} consists of terms of the form {ν} = {|λ|+|μ|−|ν|, ν_{1}, ⋯, ν_{m+1}}, where |λ| = λ_{1} + ⋯ + λ_{m}, |μ| = μ_{1} + μ_{2} and |ν| = ν_{1} + ⋯ + ν_{m+1}. As in §2, we express the multiplicity of {ν} in {λ}{μ} as c_{λμ}^{ν} and consider the generating function

F_{m,2} = ∑_{λ,μ,ν} c_{λμ}^{ν} x_{1}^{ν_{1}} ⋯ x_{m+1}^{ν_{m+1}}q_{1}^{λ_{1}} ⋯ q_{m}^{λ_{m}}r_{1}^{μ_{1}}r_{2}^{μ_{2}}.

Namely the coefficient of x_{1}^{ν_{1}} ⋯ x_{m+1}^{ν_{m+1}}q_{1}^{λ_{1}} ⋯ q_{m}^{λ_{m}}r_{1}^{μ_{1}}r_{2}^{μ_{2}} is equal to the multiplicity of {ν} in {λ}{μ}, which is not in general multiplicity-free in this case. We consider the problem expressing this generating function as a ratio of two polynomials. For small m (≤ 4) we can calculate F_{m,2} by a similar method as in the proof of Theorem 1 (see Theorem 4 below). But for general m (≥ 5) it is still undetermined, and in the next section we state several conjectures on the form of F_{m,2}.

The numerator of F_{m,2} is not so simple as that of F_{m,1}, and to express it, we first introduce one notation. Let a_{1}, a_{2}, ⋯, a_{m} be integers satisfying a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{m} ≥ 0. We define a polynomial f_{a_{1}···a_{m}} by

f_{a_{1}···a_{m}} = ∑_{b_{1},··· ,b_{m}}(x_{1}q_{1})^{b_{1}} ⋯ (x_{m}q_{m})^{b_{m}}

where the exponents b_{1} ≥ ⋯ ≥ b_{m} ≥ 0 move in the range such that the following two sets coincide:

{b_{1} − b_{2}, b_{2} − b_{3}, ⋯, b_{m−1} − b_{m}, b_{m}} = {a_{1}, a_{2}, ⋯, a_{m−1}, a_{m}}.

For example, we have

f_{211} = (x_{1}q_{1})^{4}(x_{2}q_{2})^{2}(x_{3}q_{3}) + (x_{1}q_{1})^{4}(x_{2}q_{2})^{3}(x_{3}q_{3}) + (x_{1}q_{1})^{4}(x_{2}q_{2})^{3}(x_{3}q_{3})^{2}, f_{2110} = (x_{1}q_{1})^{4}(x_{2}q_{2})^{2}(x_{3}q_{3}) + (x_{1}q_{1})^{4}(x_{2}q_{2})^{2}(x_{3}q_{3})(x_{4}q_{4}) + (x_{1}q_{1})^{4}(x_{2}q_{2})^{2}(x_{3}q_{3})^{2}(x_{4}q_{4})

+ (x_{1}q_{1})^{4}(x_{2}q_{2})^{4}(x_{3}q_{3})^{2}(x_{4}q_{4}) + (x_{1}q_{1})^{4}(x_{2}q_{2})^{3}(x_{3}q_{3})(x_{4}q_{4}) + (x_{1}q_{1})^{4}(x_{2}q_{2})^{3}(x_{3}q_{3}) + (x_{1}q_{1})^{4}(x_{2}q_{2})^{4}(x_{3}q_{3})^{3}(x_{4}q_{4}) + (x_{1}q_{1})^{4}(x_{2}q_{2})^{3}(x_{3}q_{3})^{3}(x_{4}q_{4}) + (x_{1}q_{1})^{4}(x_{2}q_{2})^{3}(x_{3}q_{3})^{2}

+ (x_{1}q_{1})^{4}(x_{2}q_{2})^{4}(x_{3}q_{3})^{3}(x_{4}q_{4})^{2} + (x_{1}q_{1})^{4}(x_{2}q_{2})^{3}(x_{3}q_{3})^{3}(x_{4}q_{4})^{2}
+ (x_{1}q_{1})^{4}(x_{2}q_{2})^{3}(x_{3}q_{3})^{2}(x_{4}q_{4})^{2}.

Note that the exponent of x_{1}q_{1} is always constant, which is equal to the sum b_{1} = ∑ a_{i}.

Theorem 4. For m = 1 ∼ 4, the generating function F_{m,2} can be expressed as a ratio of two polynomials

\frac{f(x_{1}q_{1}, ⋯, x_{m}q_{m}, r_{1}, r_{2})}{g(x_{1}, ⋯, x_{m+1}, q_{1}, ⋯, q_{m}, r_{1}, r_{2})}.

Here the numerator f is expressed as follows in terms of f_{a_{1}···a_{m}}:

m = 1 : f = 1, m = 2 : f = 1 − f_{11}r_{1}^{2}r_{2}, m = 3 : f = 1 − (f_{110} + f_{111})r_{1}^{2}r_{2} + f_{111}r_{1}^{3}r_{2} − f_{111}r_{1}^{2}r_{2}^{2} + (f_{111} + f_{211})r_{1}^{3}r_{2}^{2}

− f_{222}r_{1}^{5}r_{2}^{3},

m = 4 : f = 1 − (f_{1100} + f_{1110})r_{1}^{2}r_{2} + (f_{1110} + f_{1111})r_{1}^{3}r_{2} − f_{1111}r_{1}^{4}r_{2} − (f_{1110} + f_{1111})r_{1}^{2}r_{2}^{2} + (f_{1110} + f_{2110} + 5f_{1111} + f_{2111})r_{1}^{3}r_{2}^{2} − (f_{1111} + f_{2111})r_{1}^{4}r_{2}^{2} + (f_{1111} + f_{2111})r_{1}^{3}r_{2}^{3} − (f_{1111} + f_{2111} + f_{3111} + f_{2211})r_{1}^{4}r_{2}^{3} − (f_{2220} + f_{2221} + f_{2222} + f_{2211})r_{1}^{5}r_{2}^{3} + (f_{2221} + f_{2222})r_{1}^{6}r_{2}^{3} − (f_{2221} + f_{2222})r_{1}^{5}r_{2}^{4} + (f_{2221} + f_{3221} + 5f_{2222} + f_{3222})r_{1}^{6}r_{2}^{4} − (f_{2222} + f_{3222})r_{1}^{7}r_{2}^{4} − f_{2222}r_{1}^{5}r_{2}^{5} + (f_{2222}

+ f_{3222})r_{1}^{6}r_{2}^{5} − (f_{3222} + f_{3322})r_{1}^{7}r_{2}^{5} + f_{3333}r_{1}^{9}r_{2}^{6},

and the denominator g is given by

g(x_{1}, ⋯, x_{m+1}, q_{1}, ⋯, q_{m}, r_{1}, r_{2}) = ∏_{\{ \begin{matrix}0 ≤ i ≤ m + 1 \\ 0 ≤ j ≤ m \\ 0 ≤ k ≤ 2 \\ j + k = i + 1 \end{matrix}} (1 − x_{1} ⋯ x_{i}q_{1} ⋯ q_{j}r_{1} ⋯ r_{k})

× ∏_{2≤i+2≤j≤m}{1 − (x_{1}q_{1})^{2} ⋯ (x_{i}q_{i})^{2}(x_{i+1}q_{i+1}) ⋯ (x_{j}q_{j})r_{1}r_{2}}.

Example. In the case m = 2, the above generating function is equal to

F_{2,2} = \frac{1 − x_{1}^{2}x_{2}q_{1}^{2}q_{2}r_{1}^{2}r_{2}}{(1 − q_{1})(1 − r_{1})(1 − x_{1}q_{1}q_{2})(1 − x_{1}q_{1}r_{1})(1 − x_{1}r_{1}r_{2})(1 − x_{1}x_{2}q_{1}q_{2}r_{1})}

× (1 − x_{1}x_{2}q_{1}r_{1}r_{2})(1 − x_{1}x_{2}q_{1}q_{2}r_{1}r_{2})(1 − x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2})
= 1 + q_{1} + r_{1} + q_{1}^{2} + x_{1}q_{1}q_{2} + (x_{1} + 1)q_{1}r_{1} + · · · · · ·

+ (x_{1}^{3}x_{2} + x_{1}^{3} + x_{1}^{2}x_{2}^{2} + x_{1}^{2}x_{2}x_{3} + 2x_{1}^{2}x_{2} + x_{1}^{2} + x_{1}x_{2}x_{3} + x_{1}x_{2})q_{1}^{3}q_{2}r_{1}^{2}r_{2}

+ · · · · · · .

The coefficient of q_{1}^{3}q_{2}r_{1}^{2}r_{2} implies that {31}{21} decomposes into

{3^{2}1} + {43} + {32^{2}} + {321^{2}} + 2{421} + {52} + {41^{3}} + {51^{2}}.

In the case m = 3, we put x_{1} = ⋯ = x_{4} = 1. Then we have

f = 1 − (q_{1}^{2}q_{2} + q_{1}^{2}q_{2}q_{3} + q_{1}^{2}q_{2}^{2}q_{3} + q_{1}^{3}q_{2}^{2}q_{3})r_{1}^{2}r_{2} + q_{1}^{3}q_{2}^{2}q_{3}r_{1}^{3}r_{2} − q_{1}^{3}q_{2}^{2}q_{3}r_{1}^{2}r_{2}^{2}

+ (q_{1}^{3}q_{2}^{2}q_{3} + q_{1}^{4}q_{2}^{2}q_{3} + q_{1}^{4}q_{2}^{3}q_{3} + q_{1}^{4}q_{2}^{3}q_{3}^{2})r_{1}^{3}r_{2}^{2} − q_{1}^{6}q_{2}^{4}q_{3}^{2}r_{1}^{5}r_{2}^{3}, g = (1 − q_{1})(1 − r_{1})(1 − q_{1}q_{2})(1 − q_{1}r_{1})(1 − r_{1}r_{2})(1 − q_{1}q_{2}q_{3})(1 − q_{1}q_{2}r_{1})

× (1 − q_{1}r_{1}r_{2})(1 − q_{1}q_{2}q_{3}r_{1})(1 − q_{1}q_{2}r_{1}r_{2})^{2}(1 − q_{1}q_{2}q_{3}r_{1}r_{2})^{2}(1 − q_{1}^{2}q_{2}q_{3}r_{1}r_{2}).

And by using computers, we know that the coefficient of q_{1}^{6}q_{2}^{4}q_{3}^{2}r_{1}^{4}r_{2}^{2} in f/g is 123, which is just equal to the number of irreducible components of {642}{42}.

In the case m = 4, the numerator f stated in Theorem 4 is a little lengthy, and it actually consists of 144 monomials if we express it directly as a polynomial of x_{1}, ⋯, x_{4}, q_{1}, ⋯, q_{4}, r_{1}, r_{2}.

Before proceeding to the proof of Theorem 4, we review some relations to the previously known results. Similar generating functions are already stated in [17] and [4]. In [17; p.178] Patera and Sharp gave a formula on the generating function for SU(2) Clebsch-Gordan series:

\frac{1}{(1 − A_{1}A)(1 − A_{2}A)(1 − A_{1}A_{2})}.

This is essentially equal to the function F_{1,1} given in Theorem 1. In fact we have only to change the variables by the rule q_{1} = A_{1}A, r_{1} = A_{2}A and x_{1} = 1/A^{2}. (See also [4; p.7612].) Similarly, the generating function for SU(3) Clebsch-Gordan series stated in [17; p.178]

\frac{1}{(1 − A_{1}A)(1 − B_{1}B)(1 − A_{2}A)(1 − B_{2}B)(1 − A_{1}B_{2})(1 − B_{1}A_{2})}

× (\frac{1}{1 − A_{1}A_{2}B} + \frac{B_{1}B_{2}A}{1 − B_{1}B_{2}A})

is a special case of our generating function F_{2,2}. (See also [4; p.7634].) In this case, we substitute

q_{1} = A_{1}A, q_{2} = \frac{AB_{1}}{A_{1}} , r_{1} = A_{2}A, r_{2} = \frac{AB_{2}}{A_{2}} , x_{1} = \frac{B}{A^{2}} , x_{2} = \frac{1}{AB} , x_{3} = 0

into F_{2,2}. Then it just coincides with the above. (Actually, since the formula in [17] treats the group SU(3), it gives the multiplicity of {|λ| + |μ| − |ν|, ν_{1}, ν_{2}, 0} in {λ_{1}, λ_{2}} {μ_{1}, μ_{2}}. Our formula on the function F_{2,2} may be considered as a generalization of their result in [17] to more general ν_{3} ≠ 0 case.)

Proof of Theorem 4. We here give a proof for the case m = 3. Other cases can be treated in the same way, and we leave its examination to the readers.

We consider the Young diagram corresponding to the partition {λ_{1}, λ_{2}, λ_{3}}. First, we add a_{0}, a_{1}, a_{2}, a_{3} (a_{0} + ⋯ + a_{3} = μ_{1}) new boxes to each row, and next add b_{1}, b_{2}, b_{3}, b_{4} (b_{1} + ⋯ + b_{4} = μ_{2}) boxes to the second ∼ the fifth row under the rule of Littlewood-Richardson as follows (b_{i} boxes are represented by circles in the figure):

[figure: λ_{1}; λ_{2}; λ_{3}; a_{0}; a_{1}; a_{2}; a_{3}; b_{1}; b_{2}; b_{3}; b_{4}]

The Littlewood-Richardson rule can be expressed as the following conditions:

(3.1)  \{ \begin{matrix}a_{0} + a_{1} + a_{2} + a_{3} = μ_{1}, \\ a_{1} ≤ λ_{1} − λ_{2}, \\ a_{2} ≤ λ_{2} − λ_{3}, \\ a_{3} ≤ λ_{3}, \\ b_{1} + b_{2} + b_{3} + b_{4} = μ_{2}, \\ a_{1} + b_{1} + λ_{2} ≤ λ_{1} + a_{0}, \\ a_{2} + b_{2} + λ_{3} ≤ λ_{2} + a_{1}, \\ a_{3} + b_{3} ≤ λ_{3} + a_{2}, \\ b_{4} ≤ a_{3}, \\ b_{1} ≤ a_{0}, \\ b_{1} + b_{2} ≤ a_{0} + a_{1}, \\ b_{1} + b_{2} + b_{3} ≤ a_{0} + a_{1} + a_{2}, \\ b_{1} + b_{2} + b_{3} + b_{4} ≤ a_{0} + a_{1} + a_{2} + a_{3}. \end{matrix}

Here we put

\{ \begin{matrix}λ_{1} − λ_{2} = a_{1} + c_{1}, \\ λ_{2} − λ_{3} = a_{2} + c_{2}, \\ λ_{3} = a_{3} + c_{3}, \\ a_{3} = b_{4} + c_{4}, \\ a_{0} = b_{1} + c_{5}. \end{matrix}

Then from the above conditions (3.1) we have clearly c_{i} ≥ 0. After some calculations we know that the conditions (3.1) can be summarized to the following (3.2) and (3.3):

(3.2)  \{ \begin{matrix}λ_{1} = a_{1} + a_{2} + b_{4} + c_{1} + c_{2} + c_{3} + c_{4}, \\ λ_{2} = a_{2} + b_{4} + c_{2} + c_{3} + c_{4}, \\ λ_{3} = b_{4} + c_{3} + c_{4}, \\ μ_{1} = a_{1} + a_{2} + b_{1} + b_{4} + c_{4} + c_{5}, \\ μ_{2} = b_{1} + b_{2} + b_{3} + b_{4}, \\ a_{3} = b_{4} + c_{4}, \\ a_{0} = b_{1} + c_{5}. \end{matrix}

(3.3)  \{ \begin{matrix}b_{2} ≤ a_{1} + c_{2}, \\ b_{3} ≤ a_{2} + c_{3}, \\ b_{2} ≤ a_{1} + c_{5}, \\ b_{2} + b_{3} ≤ a_{1} + a_{2} + c_{5}. \end{matrix}

Then the generating function F_{3,2} can be expressed as follows:

F_{3,2} = ∑ x_{1}^{λ_{2}+a_{1}+b_{1}}x_{2}^{λ_{3}+a_{2}+b_{2}}x_{3}^{a_{3}+b_{3}}x_{4}^{b_{4}}q_{1}^{λ_{1}}q_{2}^{λ_{2}}q_{3}^{λ_{3}}r_{1}^{μ_{1}}r_{2}^{μ_{2}}
= ∑ x_{1}^{a_{1}+a_{2}+b_{1}+b_{4}+c_{2}+c_{3}+c_{4}}x_{2}^{a_{2}+b_{2}+b_{4}+c_{3}+c_{4}}x_{3}^{b_{3}+b_{4}+c_{4}}x_{4}^{b_{4}}q_{1}^{a_{1}+a_{2}+b_{4}+c_{1}+c_{2}+c_{3}+c_{4}}
× q_{2}^{a_{2}+b_{4}+c_{2}+c_{3}+c_{4}}q_{3}^{b_{4}+c_{3}+c_{4}}r_{1}^{a_{1}+a_{2}+b_{1}+b_{4}+c_{4}+c_{5}}r_{2}^{b_{1}+b_{2}+b_{3}+b_{4}}
= ∑ (x_{1}q_{1}r_{1})^{a_{1}}(x_{1}x_{2}q_{1}q_{2}r_{1})^{a_{2}}(x_{1}r_{1}r_{2})^{b_{1}}(x_{2}r_{2})^{b_{2}}(x_{3}r_{2})^{b_{3}}(x_{1}x_{2}x_{3}x_{4}q_{1}q_{2}q_{3}r_{1}r_{2})^{b_{4}}
× q_{1}^{c_{1}}(x_{1}q_{1}q_{2})^{c_{2}}(x_{1}x_{2}q_{1}q_{2}q_{3})^{c_{3}}(x_{1}x_{2}x_{3}q_{1}q_{2}q_{3}r_{1})^{c_{4}}r_{1}^{c_{5}},

where parameters move in the range (3.3). Since no conditions are imposed on four parameters b_{1}, b_{4}, c_{1} and c_{4}, we can first take out the term

(3.4)  \frac{1}{(1 − q_{1})(1 − x_{1}r_{1}r_{2})(1 − x_{1}x_{2}x_{3}q_{1}q_{2}q_{3}r_{1})(1 − x_{1}x_{2}x_{3}x_{4}q_{1}q_{2}q_{3}r_{1}r_{2})},

and the remaining terms are expressed as

(3.5)  ∑ (x_{1}q_{1}r_{1})^{a_{1}}(x_{1}x_{2}q_{1}q_{2}r_{1})^{a_{2}}(x_{2}r_{2})^{b_{2}}(x_{3}r_{2})^{b_{3}}(x_{1}q_{1}q_{2})^{c_{2}}(x_{1}x_{2}q_{1}q_{2}q_{3})^{c_{3}}r_{1}^{c_{5}}.

In the following, we calculate the sum (3.5) under the conditions (3.3) by dividing into five cases. (Note that all parameters are non-negative.)

(i) The case b_{2} ≤ a_{1} and b_{3} ≤ a_{2}.

In this case, the conditions (3.3) are automatically satisfied. We put a_{1} = b_{2} + d_{1}, a_{2} = b_{3} + d_{2} (d_{i} ≥ 0), and substitute them into (3.5). Then the expression (3.5) corresponding to this part is

(3.6)  ∑ (x_{1}q_{1}r_{1})^{b_{2}+d_{1}}(x_{1}x_{2}q_{1}q_{2}r_{1})^{b_{3}+d_{2}}(x_{2}r_{2})^{b_{2}}(x_{3}r_{2})^{b_{3}}(x_{1}q_{1}q_{2})^{c_{2}}(x_{1}x_{2}q_{1}q_{2}q_{3})^{c_{3}}r_{1}^{c_{5}}

= ∑ (x_{1}x_{2}q_{1}r_{1}r_{2})^{b_{2}}(x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2})^{b_{3}}(x_{1}q_{1}q_{2})^{c_{2}}(x_{1}x_{2}q_{1}q_{2}q_{3})^{c_{3}}r_{1}^{c_{5}}(x_{1}q_{1}r_{1})^{d_{1}}

× (x_{1}x_{2}q_{1}q_{2}r_{1})^{d_{2}}

= \frac{1}{(1 − r_{1})(1 − x_{1}q_{1}q_{2})(1 − x_{1}q_{1}r_{1})(1 − x_{1}x_{2}q_{1}q_{2}q_{3})(1 − x_{1}x_{2}q_{1}q_{2}r_{1})}

× (1 − x_{1}x_{2}q_{1}r_{1}r_{2})(1 − x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2}).

(ii) The case b_{2} ≤ a_{1}, b_{3} ≥ a_{2} + 1 and a_{2} + c_{5} ≥ b_{3}.

In this case we put a_{1} = b_{2} + d_{1}, b_{3} = a_{2} + d_{2} + 1. Then the remaining conditions are

\{ \begin{matrix}c_{3} ≥ d_{2} + 1, \\ c_{5} ≥ d_{2} + 1. \end{matrix}

Hence we put c_{3} = d_{2}+d_{3}+1 and c_{5} = d_{2}+d_{4}+1. Then the expression (3.5) corresponding to this part is

(3.7)  ∑ (x_{1}q_{1}r_{1})^{b_{2}+d_{1}}(x_{1}x_{2}q_{1}q_{2}r_{1})^{a_{2}}(x_{2}r_{2})^{b_{2}}(x_{3}r_{2})^{a_{2}+d_{2}+1}(x_{1}q_{1}q_{2})^{c_{2}}(x_{1}x_{2}q_{1}q_{2}q_{3})^{d_{2}+d_{3}+1}
× r_{1}^{d_{2}+d_{4}+1}

= ∑ x_{1}x_{2}x_{3}q_{1}q_{2}q_{3}r_{1}r_{2}(x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2})^{a_{2}}(x_{1}x_{2}q_{1}r_{1}r_{2})^{b_{2}}(x_{1}q_{1}q_{2})^{c_{2}}(x_{1}q_{1}r_{1})^{d_{1}}

× (x_{1}x_{2}x_{3}q_{1}q_{2}q_{3}r_{1}r_{2})^{d_{2}}(x_{1}x_{2}q_{1}q_{2}q_{3})^{d_{3}}r_{1}^{d_{4}}

= \frac{x_{1}x_{2}x_{3}q_{1}q_{2}q_{3}r_{1}r_{2}}{(1 − r_{1})(1 − x_{1}q_{1}q_{2})(1 − x_{1}q_{1}r_{1})(1 − x_{1}x_{2}q_{1}q_{2}q_{3})(1 − x_{1}x_{2}q_{1}r_{1}r_{2})}

× (1 − x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2})(1 − x_{1}x_{2}x_{3}q_{1}q_{2}q_{3}r_{1}r_{2}).

(iii) The case b_{2} ≤ a_{1}, b_{3} ≥ a_{2} + 1 and a_{2} + c_{5} + 1 ≤ b_{3}.

In this case we put a_{1} = b_{2}+d_{1}, b_{3} = a_{2}+d_{2}+1. Then from the condition a_{2}+c_{5}+1 ≤ b_{3}, we have d_{2} ≥ c_{5}. We put d_{2} = c_{5} + d_{3}. Then we have b_{3} = a_{2} + c_{5} + d_{3} + 1. The remaining conditions are

\{ \begin{matrix}c_{3} ≥ c_{5} + d_{3} + 1, \\ d_{1} ≥ d_{3} + 1. \end{matrix}

Hence we put c_{3} = c_{5}+d_{3}+d_{4}+1 and d_{1} = d_{3}+d_{5}+1. And thus we have a_{1} = b_{2}+d_{3}+d_{5}+1, b_{3} = a_{2} + c_{5} + d_{3} + 1 and c_{3} = c_{5} + d_{3} + d_{4} + 1. Then the expression (3.5) corresponding to this part is

(3.8) ∑ (x_{1}q_{1}r_{1})^{b_{2}+d_{3}+d_{5}+1}(x_{1}x_{2}q_{1}q_{2}r_{1})^{a_{2}}(x_{2}r_{2})^{b_{2}}(x_{3}r_{2})^{a_{2}+c_{5}+d_{3}+1}(x_{1}q_{1}q_{2})^{c_{2}}
× (x_{1}x_{2}q_{1}q_{2}q_{3})^{c_{5}+d_{3}+d_{4}+1}r_{1}^{c_{5}}

= ∑ x_{1}^{2}x_{2}x_{3}q_{1}^{2}q_{2}q_{3}r_{1}r_{2}(x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2})^{a_{2}}(x_{1}x_{2}q_{1}r_{1}r_{2})^{b_{2}}(x_{1}q_{1}q_{2})^{c_{2}}

× (x_{1}x_{2}x_{3}q_{1}q_{2}q_{3}r_{1}r_{2})^{c_{5}}(x_{1}^{2}x_{2}x_{3}q_{1}^{2}q_{2}q_{3}r_{1}r_{2})^{d_{3}}(x_{1}x_{2}q_{1}q_{2}q_{3})^{d_{4}}(x_{1}q_{1}r_{1})^{d_{5}}

= \frac{x_{1}^{2}x_{2}x_{3}q_{1}^{2}q_{2}q_{3}r_{1}r_{2}}{(1 − x_{1}q_{1}q_{2})(1 − x_{1}q_{1}r_{1})(1 − x_{1}x_{2}q_{1}q_{2}q_{3})(1 − x_{1}x_{2}q_{1}r_{1}r_{2})(1 − x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2})}

× (1 − x_{1}x_{2}x_{3}q_{1}q_{2}q_{3}r_{1}r_{2})(1 − x_{1}^{2}x_{2}x_{3}q_{1}^{2}q_{2}q_{3}r_{1}r_{2}).

(iv) The case b_{2} ≥ a_{1} + 1 and a_{2} ≥ b_{3}.

In this case we put b_{2} = a_{1} + d_{1} + 1, a_{2} = b_{3} + d_{2}. Then the remaining conditions are

\{ \begin{matrix}c_{2} ≥ d_{1} + 1, \\ c_{5} ≥ d_{1} + 1. \end{matrix}

And we put c_{2} = d_{1} + d_{3} + 1 and c_{5} = d_{1} + d_{4} + 1. Then the expression (3.5) corresponding to this part is

(3.9)  ∑ (x_{1}q_{1}r_{1})^{a_{1}}(x_{1}x_{2}q_{1}q_{2}r_{1})^{b_{3}+d_{2}}(x_{2}r_{2})^{a_{1}+d_{1}+1}(x_{3}r_{2})^{b_{3}}(x_{1}q_{1}q_{2})^{d_{1}+d_{3}+1}
× (x_{1}x_{2}q_{1}q_{2}q_{3})^{c_{3}}r_{1}^{d_{1}+d_{4}+1}

= ∑ x_{1}x_{2}q_{1}q_{2}r_{1}r_{2}(x_{1}x_{2}q_{1}r_{1}r_{2})^{a_{1}}(x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2})^{b_{3}}(x_{1}x_{2}q_{1}q_{2}q_{3})^{c_{3}}(x_{1}x_{2}q_{1}q_{2}r_{1}r_{2})^{d_{1}}

× (x_{1}x_{2}q_{1}q_{2}r_{1})^{d_{2}}(x_{1}q_{1}q_{2})^{d_{3}}r_{1}^{d_{4}}

= \frac{x_{1}x_{2}q_{1}q_{2}r_{1}r_{2}}{(1 − r_{1})(1 − x_{1}q_{1}q_{2})(1 − x_{1}x_{2}q_{1}q_{2}q_{3})(1 − x_{1}x_{2}q_{1}q_{2}r_{1})(1 − x_{1}x_{2}q_{1}r_{1}r_{2})}

× (1 − x_{1}x_{2}q_{1}q_{2}r_{1}r_{2})(1 − x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2}).

(v) The case b_{2} ≥ a_{1} + 1 and b_{3} ≥ a_{2} + 1.

In this case we put b_{2} = a_{1} + d_{1} + 1, b_{3} = a_{2} + d_{2} + 1. Then the remaining conditions are

\{ \begin{matrix}c_{2} ≥ d_{1} + 1, \\ c_{3} ≥ d_{2} + 1, \\ c_{5} ≥ d_{1} + 1, \\ c_{5} ≥ d_{1} + d_{2} + 2. \end{matrix}

We put c_{2} = d_{1} + d_{3} + 1, c_{3} = d_{2} + d_{4} + 1 and c_{5} = d_{1} + d_{5} + 1. Then there remains one condition d_{5} ≥ d_{2} + 1, and we put d_{5} = d_{2} + d_{6} + 1. Thus we have

\{ \begin{matrix}b_{2} = a_{1} + d_{1} + 1, \\ b_{3} = a_{2} + d_{2} + 1, \\ c_{2} = d_{1} + d_{3} + 1, \\ c_{3} = d_{2} + d_{4} + 1, \\ c_{5} = d_{1} + d_{2} + d_{6} + 2. \end{matrix}

Then the expression (3.5) corresponding to this part is

(3.10)  ∑ (x_{1}q_{1}r_{1})^{a_{1}}(x_{1}x_{2}q_{1}q_{2}r_{1})^{a_{2}}(x_{2}r_{2})^{a_{1}+d_{1}+1}(x_{3}r_{2})^{a_{2}+d_{2}+1}(x_{1}q_{1}q_{2})^{d_{1}+d_{3}+1}
× (x_{1}x_{2}q_{1}q_{2}q_{3})^{d_{2}+d_{4}+1}r_{1}^{d_{1}+d_{2}+d_{6}+2}

= ∑ x_{1}^{2}x_{2}^{2}x_{3}q_{1}^{2}q_{2}^{2}q_{3}r_{1}^{2}r_{2}^{2}(x_{1}x_{2}q_{1}r_{1}r_{2})^{a_{1}}(x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2})^{a_{2}}(x_{1}x_{2}q_{1}q_{2}r_{1}r_{2})^{d_{1}}

× (x_{1}x_{2}x_{3}q_{1}q_{2}q_{3}r_{1}r_{2})^{d_{2}}(x_{1}q_{1}q_{2})^{d_{3}}(x_{1}x_{2}q_{1}q_{2}q_{3})^{d_{4}}r_{1}^{d_{6}}

= \frac{x_{1}^{2}x_{2}^{2}x_{3}q_{1}^{2}q_{2}^{2}q_{3}r_{1}^{2}r_{2}^{2}}{(1 − r_{1})(1 − x_{1}q_{1}q_{2})(1 − x_{1}x_{2}q_{1}q_{2}q_{3})(1 − x_{1}x_{2}q_{1}r_{1}r_{2})(1 − x_{1}x_{2}q_{1}q_{2}r_{1}r_{2})}

× (1 − x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2})(1 − x_{1}x_{2}x_{3}q_{1}q_{2}q_{3}r_{1}r_{2}).

Finally we add the expressions (3.6) ∼ (3.10) and multiply (3.4). Then after some calculations we have F_{3,2} = f/g where

f = 1 − (x_{1}^{2}x_{2}q_{1}^{2}q_{2} + x_{1}^{2}x_{2}x_{3}q_{1}^{2}q_{2}q_{3} + x_{1}^{2}x_{2}^{2}x_{3}q_{1}^{2}q_{2}^{2}q_{3} + x_{1}^{3}x_{2}^{2}x_{3}q_{1}^{3}q_{2}^{2}q_{3})r_{1}^{2}r_{2}
+ x_{1}^{3}x_{2}^{2}x_{3}q_{1}^{3}q_{2}^{2}q_{3}r_{1}^{3}r_{2} − x_{1}^{3}x_{2}^{2}x_{3}q_{1}^{3}q_{2}^{2}q_{3}r_{1}^{2}r_{2}^{2} + (x_{1}^{3}x_{2}^{2}x_{3}q_{1}^{3}q_{2}^{2}q_{3}
+ x_{1}^{4}x_{2}^{2}x_{3}q_{1}^{4}q_{2}^{2}q_{3} + x_{1}^{4}x_{2}^{3}x_{3}q_{1}^{4}q_{2}^{3}q_{3} + x_{1}^{4}x_{2}^{3}x_{3}^{2}q_{1}^{4}q_{2}^{3}q_{3}^{2})r_{1}^{3}r_{2}^{2}
− x_{1}^{6}x_{2}^{4}x_{3}^{2}q_{1}^{6}q_{2}^{4}q_{3}^{2}r_{1}^{5}r_{2}^{3},
g = (1 − q_{1})(1 − r_{1})(1 − x_{1}q_{1}q_{2})(1 − x_{1}q_{1}r_{1})(1 − x_{1}r_{1}r_{2})(1 − x_{1}x_{2}q_{1}q_{2}q_{3})
× (1 − x_{1}x_{2}q_{1}q_{2}r_{1})(1 − x_{1}x_{2}q_{1}r_{1}r_{2})(1 − x_{1}x_{2}x_{3}q_{1}q_{2}q_{3}r_{1})(1 − x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2})
× (1 − x_{1}x_{2}x_{3}x_{4}q_{1}q_{2}q_{3}r_{1}r_{2})(1 − x_{1}x_{2}q_{1}q_{2}r_{1}r_{2})(1 − x_{1}x_{2}x_{3}q_{1}q_{2}q_{3}r_{1}r_{2})
× (1 − x_{1}^{2}x_{2}x_{3}q_{1}^{2}q_{2}q_{3}r_{1}r_{2}).

By definition we have

f_{110} = x_{1}^{2}x_{2}q_{1}^{2}q_{2} + x_{1}^{2}x_{2}x_{3}q_{1}^{2}q_{2}q_{3} + x_{1}^{2}x_{2}^{2}x_{3}q_{1}^{2}q_{2}^{2}q_{3},
f_{111} = x_{1}^{3}x_{2}^{2}x_{3}q_{1}^{3}q_{2}^{2}q_{3},
f_{211} = x_{1}^{4}x_{2}^{2}x_{3}q_{1}^{4}q_{2}^{2}q_{3} + x_{1}^{4}x_{2}^{3}x_{3}q_{1}^{4}q_{2}^{3}q_{3} + x_{1}^{4}x_{2}^{3}x_{3}^{2}q_{1}^{4}q_{2}^{3}q_{3}^{2},
f_{222} = x_{1}^{6}x_{2}^{4}x_{3}^{2}q_{1}^{6}q_{2}^{4}q_{3}^{2},

and hence

f = 1 − (f_{110} + f_{111})r_{1}^{2}r_{2} + f_{111}r_{1}^{3}r_{2} − f_{111}r_{1}^{2}r_{2}^{2} + (f_{111} + f_{211})r_{1}^{3}r_{2}^{2} − f_{222}r_{1}^{5}r_{2}^{3}.

Therefore we obtain the desired expression for the case m = 3. q.e.d.

Concerning the decomposition of {λ_{1}, λ_{2}}{μ_{1}, μ_{2}}, an explicit decomposition formula is already given in Remmel-Whitehead [22]. As a corollary of Theorem 4, we give the following quite simple new decomposition formula that resembles the one in Corollary 3.

14 Y. Agaoka

Corollary 5. The following decomposition formula holds: {λ_{1}, λ_{2}}{μ_{1}, μ_{2}} =

∑_{\{ \begin{matrix}a + d + g = λ_{1} − λ_{2} \\ c + f + h + i = λ_{2} \\ b + d + f = μ_{1} − μ_{2} \\ e + g + h + i = μ_{2} \\ a, b, c, d, e, f, g, h, i ≥ 0 \\ dh = 0 \end{matrix}} \begin{matrix}[ (a + b)\{1, 0, 0, 0\} + (c + d + e)\{1, 1, 0, 0\} + (f + g)\{1, 1, 1, 0\} \\ + h\{2, 1, 1, 0\} + i\{1, 1, 1, 1\} ] . \end{matrix}

Example. We consider the case {32}{21}. In this case the set of integers satisfying the conditions

\{ \begin{matrix}a + d + g = 3 − 2 = 1, \\ c + f + h + i = 2, \\ b + d + f = 2 − 1 = 1, \\ e + g + h + i = 1, \\ a, b, c, d, e, f, g, h, i ≥ 0, \\ dh = 0 \end{matrix}

is given as follows:

\begin{matrix}a & b & c & d & e & f & g & h & i & \{ν\} \\ 1 & 1 & 2 & 0 & 1 & 0 & 0 & 0 & 0 & \{53\} \\ 0 & 1 & 2 & 0 & 0 & 0 & 1 & 0 & 0 & \{431\} \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & \{521\} \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & \{421^{2}\} \\ 0 & 0 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & \{4^{2}\} \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & \{3^{2}1^{2}\} \\ 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & \{3^{2}2\} \\ 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & \{431\} \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & \{42^{2}\} \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & \{32^{2}1\} \end{matrix}

Hence we have the decomposition {32}{21} = {53} + {521} + {4^{2}} + 2{431} + {42^{2}} + {421^{2}} + {3^{2}2} + {3^{2}1^{2}} + {32^{2}1}.

Proof of Corollary 5. We follow a similar method as in the proof of Corollary 3. First, we have

F_{2,2} = \frac{1 − x_{1}^{2}x_{2}q_{1}^{2}q_{2}r_{1}^{2}r_{2}}{(1 − q_{1})(1 − r_{1})(1 − x_{1}q_{1}q_{2})(1 − x_{1}q_{1}r_{1})(1 − x_{1}r_{1}r_{2})(1 − x_{1}x_{2}q_{1}q_{2}r_{1})}

× (1 − x_{1}x_{2}q_{1}r_{1}r_{2})(1 − x_{1}x_{2}q_{1}q_{2}r_{1}r_{2})(1 − x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2})
= ∑ q_{1}^{a}r_{1}^{b}(x_{1}q_{1}q_{2})^{c}(x_{1}q_{1}r_{1})^{d}(x_{1}r_{1}r_{2})^{e}(x_{1}x_{2}q_{1}q_{2}r_{1})^{f}(x_{1}x_{2}q_{1}r_{1}r_{2})^{g}
× (x_{1}x_{2}q_{1}q_{2}r_{1}r_{2})^{h}(x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2})^{i}
− ∑ q_{1}^{a}r_{1}^{b}(x_{1}q_{1}q_{2})^{c}(x_{1}q_{1}r_{1})^{d}(x_{1}r_{1}r_{2})^{e}(x_{1}x_{2}q_{1}q_{2}r_{1})^{f}(x_{1}x_{2}q_{1}r_{1}r_{2})^{g}
× (x_{1}x_{2}q_{1}q_{2}r_{1}r_{2})^{h}(x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2})^{i}(x_{1}^{2}x_{2}q_{1}^{2}q_{2}r_{1}^{2}r_{2})

= ∑ x_{1}^{c+d+e+f+g+h+i}x_{2}^{f+g+h+i}x_{3}^{i}q_{1}^{a+c+d+f+g+h+i}q_{2}^{c+f+h+i}r_{1}^{b+d+e+f+g+h+i}r_{2}^{e+g+h+i}

− ∑ x_{1}^{c+d+e+f+g+h+i+2}x_{2}^{f+g+h+i+1}x_{3}^{i}q_{1}^{a+c+d+f+g+h+i+2}q_{2}^{c+f+h+i+1}
× r_{1}^{b+d+e+f+g+h+i+2}r_{2}^{e+g+h+i+1}.

The first line in the last expression corresponds to the partition

{∗, c + d + e + f + g + h + i, f + g + h + i, i}

in {λ_{1}, λ_{2}}{μ_{1}, μ_{2}} = {a+c+d+f +g+h+i, c+f +h+i}{b+d+e+f +g+h+i, e+g+h+i}. It is easy to see that the above ∗ part is equal to a + b + c + d + e + f + g + 2h + i. Hence, the first line is equal to

∑_{\{ \begin{matrix}a + d + g = λ_{1} − λ_{2} \\ c + f + h + i = λ_{2} \\ b + d + f = μ_{1} − μ_{2} \\ e + g + h + i = μ_{2} \\ a, b, c, d, e, f, g, h, i ≥ 0 \end{matrix}} \begin{matrix}[ (a + b)\{1, 0, 0, 0\} + (c + d + e)\{1, 1, 0, 0\} + (f + g)\{1, 1, 1, 0\} \\ + h\{2, 1, 1, 0\} + i\{1, 1, 1, 1\} ] . \end{matrix}

In the similar way, the second line corresponds to the partitions

{a + b + c + d + e + f + g + 2h + i + 3, c + d + e + f + g + h + i + 2, f + g + h + i + 1, i}

in {λ_{1}, λ_{2}}{μ_{1}, μ_{2}} = {a + c + d + f + g + h + i + 2, c + f + h + i + 1}{b + d + e + f + g + h + i + 2, e + g + h + i + 1}. In this case, the above partition can be expressed as

(a + b){1, 0, 0, 0} + (c + d + e){1, 1, 0, 0} + (f + g){1, 1, 1, 0}
+ h{2, 1, 1, 0} + i{1, 1, 1, 1} + {3, 2, 1, 0}
= (a + b){1, 0, 0, 0} + (c + (d + 1) + e){1, 1, 0, 0} + (f + g){1, 1, 1, 0}
+ (h + 1){2, 1, 1, 0} + i{1, 1, 1, 1}.

In addition, we have

\{ \begin{matrix}a + (d + 1) + g = λ_{1} − λ_{2}, \\ c + f + (h + 1) + i = λ_{2}, \\ b + (d + 1) + f = μ_{1} − μ_{2}, \\ e + g + (h + 1) + i = μ_{2}. \end{matrix}

Hence, by changing the variables d, h, we know that the second line is equal to

∑_{\{ \begin{matrix}a + d + g = λ_{1} − λ_{2} \\ c + f + h + i = λ_{2} \\ b + d + f = μ_{1} − μ_{2} \\ e + g + h + i = μ_{2} \\ a, b, c, e, f, g, i ≥ 0, d, h ≥ 1 \end{matrix}}\begin{matrix}[ (a + b)\{1, 0, 0, 0\} + (c + d + e)\{1, 1, 0, 0\} + (f + g)\{1, 1, 1, 0\} \\ + h\{2, 1, 1, 0\} + i\{1, 1, 1, 1\} ] . \end{matrix}

Subtracting this expression from the first line, we obtain the desired result. q.e.d.

Remark. From Corollary 5, we know that the number of irreducible components of {λ_{1}, λ_{2}}{μ_{1}, μ_{2}} is equal to the number of lattice points in R^{5} with coordinate (d, f, g, h, i) satisfying the following conditions:

\{ \begin{matrix}d + g ≤ λ_{1} − λ_{2}, \\ f + h + i ≤ λ_{2}, \\ d + f ≤ μ_{1} − μ_{2}, \\ g + h + i ≤ μ_{2}, \\ d, f, g, h, i ≥ 0, \\ dh = 0. \end{matrix}

In general it is hard to express this number explicitly as a polynomial of λ_{i} and μ_{i}. As one example, in case 0 ≤ μ_{2} ≤ λ_{2} ≤ μ_{1} − μ_{2} ≤ μ_{1} ≤ λ_{1} − λ_{2}, we can directly verify that the number of irreducible components of {λ_{1}, λ_{2}}{μ_{1}, μ_{2}} is equal to

(λ_{2}μ_{1} + μ_{1} − μ_{2} + 1)\binom{μ_{2} + 2}{2} − \binom{λ_{2}}{2}\binom{μ_{2} + 2}{2} − (λ_{2} + μ_{1})\binom{μ_{2} + 2}{3} + \binom{μ_{2} + 2}{4}.

4. Conjectures.

Generation functions stated in Theorem 4 possess several properties. In this section we summarize these properties in the form of conjectures on F_{m,2} (Conjecture 1), and in addition, give the explicit conjectural expression of the quite lengthy next generating function F_{5,2} (Conjecture 2). We use the same notations as in the previous section.

Conjecture 1. F_{m,2} can be expressed as a ratio of two polynomials:

\frac{f(x_{1}q_{1}, ⋯, x_{m}q_{m}, r_{1}, r_{2})}{g(x_{1}, ⋯, x_{m+1}, q_{1}, ⋯, q_{m}, r_{1}, r_{2})}.

Here, the denominator g is given by

g(x_{1}, ⋯, x_{m+1}, q_{1}, ⋯, q_{m}, r_{1}, r_{2}) = ∏_{\{ \begin{matrix}0 ≤ i ≤ m + 1 \\ 0 ≤ j ≤ m \\ 0 ≤ k ≤ 2 \\ j + k = i + 1 \end{matrix}} (1 − x_{1} ⋯ x_{i}q_{1} ⋯ q_{j}r_{1} ⋯ r_{k})

× ∏_{2≤i+2≤j≤m}{1 − (x_{1}q_{1})^{2} ⋯ (x_{i}q_{i})^{2}(x_{i+1}q_{i+1}) ⋯ (x_{j}q_{j})r_{1}r_{2}}

and the numerator f can be expressed as a linear combination of f_{a_{1}···a_{m}}r_{1}^{p}r_{2}^{q} (a_{1} ≥ ⋯ ≥ a_{m} ≥ 0 and p ≥ q ≥ 0) with integer coefficients. The numerator f satisfies the following properties:

(i) If the term f_{a_{1}···a_{m}}r_{1}^{p}r_{2}^{q} appears in f, then the exponent (p, q) moves in the range

0 ≤ p ≤ \frac{(m−1)(m+2)}{2}, 0 ≤ q ≤ \frac{m(m−1)}{2}, p − q ≤ m − 1,

max \{p, p + q − (m − 1)\} ≤ ∑_{i=1}^{m} a_{i} ≤ min \{p + q, p + \frac{(m−1)(m−2)}{2}\}.

(ii) The coefficients of r_{2}^{0} and r_{2}^{1} in f are given by

1 − ∑_{k=2}^{m−1}(−1)^{k}(f_{^{1 ⋯ 1}_{k} ^{0 ⋯ 0}_{m−k}} + f_{^{1 ⋯ 1}_{k+1} ^{0 ⋯ 0}_{m−k−1}}) r_{1}^{k}r_{2} − (−1)^{m}f^{1 ⋯ 1}_{m} r_{1}^{m}r_{2}.

(iii) The coefficient of r_{1}^{2}r_{2}^{2} in f is given by

−(f_{111 ^{0 ⋯ 0}_{m−3}} + f_{1111 ^{0 ⋯ 0}_{m−4}}) r_{1}^{2}r_{2}^{2}.

(iv) The coefficient of r_{1}^{\frac{(m−1)(m+2)}{2}} r_{2}^{\frac{m(m−1)}{2}} , which is the last term of f, is given by

(−1)^{\frac{m(m−1)}{2}} f^{m − 1, ⋯, m − 1}_{m} r_{1}^{\frac{(m−1)(m+2)}{2}} r_{2}^{\frac{m(m−1)}{2}} .

(v) The following reciprocal property holds:

f(x_{1}q_{1}, ⋯, x_{m}q_{m}, r_{1}, r_{2})

= (−1)^{\frac{m(m−1)}{2}} (x_{1}q_{1})^{m(m−1)}(x_{2}q_{2})^{(m−1)^{2}} ⋯ (x_{m−1}q_{m−1})^{2(m−1)}(x_{m}q_{m})^{m−1}

× r_{1}^{\frac{(m−1)(m+2)}{2}} r_{2}^{\frac{m(m−1)}{2}} f (\frac{1}{x_{1}q_{1}} , ⋯, \frac{1}{x_{m}q_{m}} , \frac{1}{r_{1}} , \frac{1}{r_{2}} ) .

(vi) If we set x_{m}q_{m} = 0 in f, then the polynomial f(x_{1}q_{1}, ⋯, x_{m−1}q_{m−1}, 0, r_{1}, r_{2}) gives the numerator of F_{m−1,2}.

Remark that the parameter x_{m+1} appears only in the denominator of F_{m,2}. If we put r_{2} = 0 in F_{m,2}, then we just obtain the generating function F_{m,1} appeared in Theorem 1. Unfortunately, the numerator f cannot be uniquely characterized by only the above properties.

It is easy to see that the reciprocal property (v) in Conjecture 1 is equivalent to say that the term kf_{a_{1}···a_{m}}r_{1}^{p}r_{2}^{q} (k ∈ Z) appears in the numerator f if and only if the term (−1)^{\frac{m(m−1)}{2}} kf_{(m−1)−a_{m},(m−1)−a_{m−1},··· ,(m−1)−a_{1}}r_{1}^{\frac{(m−1)(m+2)}{2} −p} r_{2}^{\frac{m(m−1)}{2} −q} appears in f. (Remark that f_{0···0} = 1.)

We can easily see that Conjecture 1 actually holds for the case m ≤ 4. For the next case m = 5, it seems hard to obtain the explicit formula by the method

stated in the proof of Theorem 4. But by using computers, we now arrive at the following conjecture.

Conjecture 2. In case m = 5, the numerator of the generating function is given as follows: f(x_{1}q_{1}, ⋯, x_{5}q_{5}, r_{1}, r_{2}) = 1 − (f_{11000} + f_{11100})r_{1}^{2}r_{2} + (f_{11100} + f_{11110})r_{1}^{3}r_{2} − (f_{11110} + f_{11111})r_{1}^{4}r_{2} + f_{11111}r_{1}^{5}r_{2} − (f_{11100} + f_{11110})r_{1}^{2}r_{2}^{2} + (f_{11100} + f_{21100} + 5f_{11110} + f_{21110} + 5f_{11111})r_{1}^{3}r_{2}^{2} − (f_{11110} + f_{21110} + 6f_{11111} + f_{21111})r_{1}^{4}r_{2}^{2} + (f_{11111} + f_{21111})r_{1}^{5}r_{2}^{2} + (f_{11110} + f_{21110} + 5f_{11111} + f_{21111})r_{1}^{3}r_{2}^{3} − (f_{11110} + f_{21110} + 6f_{11111} + f_{31110} + f_{22110} + 6f_{21111} + f_{31111} + f_{22111})r_{1}^{4}r_{2}^{3} + (f_{11111} − f_{22200} − f_{22110} − f_{21111} + f_{31111} − f_{22210} − f_{22111} − f_{22220} − f_{22211})r_{1}^{5}r_{2}^{3} + (f_{22210} + 2f_{22111} + f_{22220} + 2f_{22211} + f_{22221})r_{1}^{6}r_{2}^{3} − (f_{22211} + f_{22221}) r_{1}^{7}r_{2}^{3} − (f_{11111} + f_{21111} + f_{31111} + f_{22111})r_{1}^{4}r_{2}^{4} + (f_{11111} + f_{21111} + f_{31111} − f_{22210} − f_{22111} + f_{41111} + f_{32111} − f_{22220} − f_{22211} − f_{22221})r_{1}^{5}r_{2}^{4} + (f_{22210} + 2f_{22111} + f_{32210} + 2f_{32111} + 5f_{22220} + 10f_{22211} + f_{32220} + 2f_{32211} + 10f_{22221} + f_{32221} + 5f_{22222})r_{1}^{6}r_{2}^{4} − (f_{22220} + 2f_{22211} + f_{32220} + 2f_{32211} + 10f_{22221} + 2f_{32221} + 10f_{22222} + f_{32222})r_{1}^{7}r_{2}^{4} + (f_{22221} + f_{32221} + 5f_{22222} + f_{32222})r_{1}^{8}r_{2}^{4} − (f_{22220} + f_{22211} + f_{22221} + f_{22222})r_{1}^{5}r_{2}^{5} + (f_{22220} + 2f_{22211} + f_{32220} + 2f_{32211} + 10f_{22221} + 2f_{32221} + 10f_{22222} + f_{32222})r_{1}^{6}r_{2}^{5} − (f_{22211} + f_{32220} + 2f_{32211} + 10f_{22221} + f_{42211} + f_{33220} + 2f_{33211} + 10f_{32221} + 30f_{22222} + f_{42221} + 2f_{33221} + 10f_{32222} + f_{33222})r_{1}^{7}r_{2}^{5} + (f_{22221} + 10f_{22222} + 2f_{32221} + 10f_{32222} + 2f_{33221} + f_{42221} + 2f_{33222} + f_{42222})r_{1}^{8}r_{2}^{5} − (f_{42222} + f_{33222} + f_{32222} + f_{22222})r_{1}^{9}r_{2}^{5} + (f_{32222} + f_{32221} + 5f_{22222} + f_{22221})r_{1}^{6}r_{2}^{6} − (f_{42222} + 2f_{33222} + f_{42221} + 2f_{33221} + 10f_{32222} + 2f_{32221} + 10f_{22222} + f_{22221})r_{1}^{7}r_{2}^{6} + (f_{43222} + 2f_{33322} + f_{43221} + 2f_{33321} + 5f_{42222} + 10f_{33222} + f_{42221} + 2f_{33221} + 10f_{32222} + f_{32221} + 5f_{22222})r_{1}^{8}r_{2}^{6} + (f_{33333} + f_{33332} + f_{33331} − f_{43222} − f_{33322} + f_{33330} + f_{33321} − f_{42222} − f_{33222} − f_{32222})r_{1}^{9}r_{2}^{6} − (f_{33333} + f_{33332} + f_{33331} + f_{33322})r_{1}^{10}r_{2}^{6} − (f_{33222} + f_{32222})r_{1}^{7}r_{2}^{7} + (f_{43222} + 2f_{33322} + f_{42222} + 2f_{33222} + f_{32222})r_{1}^{8}r_{2}^{7} + (f_{33333} − f_{44222} − f_{43322} − f_{33332} + f_{33331} − f_{43222} − f_{33322} − f_{42222} − f_{33222})r_{1}^{9}r_{2}^{7} − (f_{43333} + f_{43332} + 6f_{33333} + f_{43331} + f_{43322} + 6f_{33332} + f_{33331} + f_{33322})r_{1}^{10}r_{2}^{7} + (f_{43333} + f_{43332} + 5f_{33333} + f_{33332})r_{1}^{11}r_{2}^{7} + (f_{33333} + f_{33332}) r_{1}^{9}r_{2}^{8} − (f_{43333} + f_{43332} + 6f_{33333} + f_{33332})r_{1}^{10}r_{2}^{8} + (f_{44333} + f_{44332} + 5f_{43333} + f_{43332} + 5f_{33333})r_{1}^{11}r_{2}^{8} − (f_{44333} + f_{43333})r_{1}^{12}r_{2}^{8} − f_{33333}r_{1}^{9}r_{2}^{9} − (f_{43333} + f_{33333})r_{1}^{10}r_{2}^{9} + (f_{44333} + f_{43333})r_{1}^{11}r_{2}^{9} − (f_{44433} + f_{44333})r_{1}^{12}r_{2}^{9} + f_{44444}r_{1}^{14}r_{2}^{10}.

This numerator actually consists of 3700 monomials if we express it as a polynomial of x_{i}, q_{i} and r_{i}. The coefficients appeared in this numerator are limited to 1, 2, 5, 6, 10, 30, and they may have some combinatorial meaning. It can be directly verified that the numerator f in this conjecture satisfies all properties stated in Conjecture 1.

5. Littlewood-Richardson polynomials.

We fix three partitions {λ}, {μ}, {ν} such that |λ| + |μ| = |ν|. Then it is known that for non-negative integers N, the value c_{Nλ Nμ}^{Nν} can be expressed as a polynomial of N, and this polynomial is now called the Littlewood-Richardson polynomial (cf. [9], [18]). By using the method of hives (cf. Buch [6], or Knutson-Tao [12]), we can easily see that the degree of the Littlewood-Richardson polynomial is at most (m − 1)(n − 1) for the case {λ} = {λ_{1}, ⋯, λ_{m}} and {μ} = {μ_{1}, ⋯, μ_{n}}. (In fact in this case, the Littlewood-Richardson coefficient c_{λμ}^{ν} is equal to the number of lattice points in some (m − 1)(n − 1)- dimensional convex polytope whose boundaries are determined by the conditions on hives.) In this section, we give a generating function of the Littlewood-Richardson polynomial for the case {λ} = {λ_{1}, λ_{2}} and {μ} = {μ_{1}, μ_{2}}, as an application of Theorem 4.

First we define a polynomial f_{λμ}^{ν}(N) by

f_{λμ}^{ν}(N) = \{ \begin{matrix}c_{Nλ Nμ}^{Nν} & (c_{λμ}^{ν} ≠ 0), \\ 0 & (c_{λμ}^{ν} = 0). \end{matrix}

And next, define a generating function LR_{2,2}(N) by

LR_{2,2}(N) = ∑_{\begin{matrix}\{λ\} = \{λ_{1}, λ_{2}\} \\ \{μ\} = \{μ_{1}, μ_{2}\} \\ \{ν\} = \{ν_{1}, ν_{2}, ν_{3}, ν_{4}\} \\ |λ| + |μ| = |ν| \end{matrix}} f_{λμ}^{ν}(N) x_{1}^{ν_{2}}x_{2}^{ν_{3}}x_{3}^{ν_{4}}q_{1}^{λ_{1}}q_{2}^{λ_{2}}r_{1}^{μ_{1}}r_{2}^{μ_{2}}.

Namely, the coefficient of x_{1}^{ν_{2}}x_{2}^{ν_{3}}x_{3}^{ν_{4}}q_{1}^{λ_{1}}q_{2}^{λ_{2}}r_{1}^{μ_{1}}r_{2}^{μ_{2}} in LR_{2,2}(N) gives the Littlewood-Richardson polynomial for {λ} = {λ_{1}, λ_{2}}, {μ} = {μ_{1}, μ_{2}} and {ν} = {ν_{1}, ⋯, ν_{4}}. Then we have the following theorem.

Theorem 6. The generating function LR_{2,2}(N) is given by the following form:

(1 − x_{1}^{2}x_{2}q_{1}^{2}q_{2}r_{1}^{2}r_{2})(1 + x_{1}^{2}x_{2}q_{1}^{2}q_{2}r_{1}^{2}r_{2}(N − 1))
(1 − q_{1})(1 − r_{1})(1 − x_{1}q_{1}q_{2})(1 − x_{1}q_{1}r_{1})(1 − x_{1}r_{1}r_{2})(1 − x_{1}x_{2}q_{1}q_{2}r_{1})
× (1 − x_{1}x_{2}q_{1}r_{1}r_{2})(1 − x_{1}x_{2}q_{1}q_{2}r_{1}r_{2})(1 − x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2}).

As an example, by using computers we know that the coefficient of x_{1}^{7}x_{2}^{3}q_{1}^{7}q_{2}^{4}r_{1}^{6}r_{2}^{3} in LR_{2,2}(N) is 3N + 1, which gives the Littlewood-Richardson polynomial for the case {λ} = {74}, {μ} = {63} and {ν} = {10, 73}. Note that if we put N = 1 into LR_{2,2}(N), then we just obtain the generating function F_{2,2} in Theorem 4.

Since the degree of the Littlewood-Richardson polynomial is at most 1 for the case {λ} = {λ_{1}, λ_{2}} and {μ} = {μ_{1}, μ_{2}}, it suffices to know the function LR_{2,2}(0) in order to prove Theorem 6. In fact, using the function F_{2,2}, LR_{2,2}(N) can be expressed as

LR_{2,2}(N) = (F_{2,2} − LR_{2,2}(0))N + LR_{2,2}(0).

The generating function LR_{2,2}(0) may be considered as a support function of c_{λμ}^{ν}. Namely,

c_{λμ}^{ν} > 0 ⟺ the term x_{1}^{ν_{2}}x_{2}^{ν_{3}}x_{3}^{ν_{4}}q_{1}^{λ_{1}}q_{2}^{λ_{2}}r_{1}^{μ_{1}}r_{2}^{μ_{2}} appears in LR_{2,2}(0).

(By definition, the coefficient of each term in LR_{2,2}(0) is always 1.) The generating function LR_{2,2}(0) is given in the following form. Theorem 6 follows immediately from this proposition.

Proposition 7.

LR_{2,2}(0) = \frac{(1 − x_{1}^{2}x_{2}q_{1}^{2}q_{2}r_{1}^{2}r_{2})^{2}}{(1 − q_{1})(1 − r_{1})(1 − x_{1}q_{1}q_{2})(1 − x_{1}q_{1}r_{1})(1 − x_{1}r_{1}r_{2})(1 − x_{1}x_{2}q_{1}q_{2}r_{1})}

× (1 − x_{1}x_{2}q_{1}r_{1}r_{2})(1 − x_{1}x_{2}q_{1}q_{2}r_{1}r_{2})(1 − x_{1}x_{2}x_{3}q_{1}q_{2}r_{1}r_{2}).

It is surprising that all coefficients in this (somewhat complicated) formal power series are 1. If we put x_{1} = x_{2} = x_{3} = 1 into the above LR_{2,2}(0), we obtain the generating function expressing the number of different “types” of partitions appearing in the tensor product {λ_{1}, λ_{2}}{μ_{1}, μ_{2}}. For example, it is expanded as

1 + q_{1} + r_{1} + q_{1}^{2} + q_{1}q_{2} + r_{1}^{2} + r_{1}r_{2} + 2q_{1}r_{1} + · · · · · · + 13q_{1}^{5}q_{2}r_{1}^{3}r_{2}^{2} + · · · · · · ,

and hence we know that 13 different types of partitions appear in the product {51}{32}.

We can prove Proposition 7 by a similar method stated in the proof of Theorem 4, and we here state only the outline of the proof. First, by using the Littlewood-Richardson rule (or hives), we can easily show that c_{λ μ}^{ν} > 0 if and only if

|λ| + |μ| = |ν|,
ν_{4} ≤ min {λ_{2}, μ_{2}}, max {λ_{2}, μ_{2}} ≤ ν_{2},
ν_{3} ≤ min {λ_{1}, μ_{1}}, max {λ_{1}, μ_{1}} ≤ ν_{1},
ν_{3} + ν_{4} ≤ λ_{2} + μ_{2} ≤ ν_{2} + ν_{3} ≤ λ_{1} + μ_{1},
ν_{2} + ν_{4} ≤ min {λ_{1} + μ_{2}, λ_{2} + μ_{1}}.

Under these conditions, we calculate the sum

∑_{λ, μ, ν} x_{1}^{ν_{2}}x_{2}^{ν_{3}}x_{3}^{ν_{4}}q_{1}^{λ_{1}}q_{2}^{λ_{2}}r_{1}^{μ_{1}}r_{2}^{μ_{2}}.

Then after many case by case examinations, we can finally obtain the expression stated in Proposition 7.

Finally, we give two applications of our formula on generating functions. The first one concerns the multiplicity-free product. In [26] Stembridge gave a classification of the pair (λ, μ) such that the product {λ}{μ} is multiplicity-free. We here give another proof of this result for the case {λ} = {λ_{1}, λ_{2}} and {μ} = {μ_{1}, μ_{2}}, by using our generating function F_{2,2} and LR_{2,2}(0).

Theorem 8 ([26]). The product {λ_{1}, λ_{2}}{μ_{1}, μ_{2}} is multiplicity-free if and only if {λ_{1}, λ_{2}} or {μ_{1}, μ_{2}} forms a rectangle.

Proof. It is clear that the product {λ_{1}, λ_{2}}{μ_{1}, μ_{2}} is multiplicity-free if and only if the coefficient of x_{1}^{ν_{2}}x_{2}^{ν_{3}}x_{3}^{ν_{4}}q_{1}^{λ_{1}}q_{2}^{λ_{2}}r_{1}^{μ_{1}}r_{2}^{μ_{2}} in F_{2,2} is 1 for any {ν}. Since the coefficient of each term in LR_{2,2}(0) = (1 − x_{1}^{2}x_{2}q_{1}^{2}q_{2}r_{1}^{2}r_{2})F_{2,2} is always 1, the above condition is equivalent to say that the coefficient of q_{1}^{λ_{1}}q_{2}^{λ_{2}}r_{1}^{μ_{1}}r_{2}^{μ_{2}} in F_{2,2} and in (1 − x_{1}^{2}x_{2}q_{1}^{2}q_{2}r_{1}^{2}r_{2})F_{2,2} coincides. If the term containing q_{1}^{λ_{1}−2}q_{2}^{λ_{2}−1}r_{1}^{μ_{1}−2}r_{2}^{μ_{2}−1} appears in F_{2,2}, then q_{1}^{λ_{1}}q_{2}^{λ_{2}}r_{1}^{μ_{1}}r_{2}^{μ_{2}} does not possess this property. And the converse is also true. Clearly the term containing q_{1}^{λ_{1}−2}q_{2}^{λ_{2}−1}r_{1}^{μ_{1}−2}r_{2}^{μ_{2}−1} does not appear in F_{2,2} if and only if {λ_{1}, λ_{2}} or {μ_{1}, μ_{2}} is a rectangle, and thus we complete the proof of Theorem 8. q.e.d.

Next, as the second application, we give the explicit expression of the Littlewood-Richardson polynomial. We can easily see that the generating function LR_{2,2}(N) also can be expressed as

LR_{2,2}(N) = LR_{2,2}(0) + x_{1}^{2}x_{2}q_{1}^{2}q_{2}r_{1}^{2}r_{2}F_{2,2} × N.

By using this equality, we can easily show the following theorem.

Theorem 9. Assume that the partitions {λ} = {λ_{1}, λ_{2}}, {μ} = {μ_{1}, μ_{2}}, {ν} = \{ν_{1}, ⋯, ν_{4}\} satisfy c_{λ μ}^{ν} ≠ 0, and we put \{\overline{λ}\} = \{λ_{1} − 2, λ_{2} − 1\}, \{\overline{μ}\} = \{μ_{1} − 2, μ_{2} − 1\}, \{\overline{ν}\} = \{ν_{1}−3, ν_{2}−2, ν_{3}−1, ν_{4}\}. Then the Littlewood-Richardson polynomial corresponding to the partitions \{λ\}, \{μ\} and \{ν\} is given by c\overline{ν} ; \overline{λ} \overline{μ}N + 1. (If \{\overline{λ}\}, \{\overline{μ}\} or \{\overline{ν}\} does not form a partition, we consider c\overline{ν} ; \overline{λ} \overline{μ} = 0.)

The essence of the proof is almost same to that of Theorem 8, and we leave the proof to the readers.

In contrast to the above theorem, it is in general hard to express the Littlewood-Richardson polynomial as an explicit function of {λ}, {μ} and {ν} because it requires quite complicated case by case verification. For example, assume that the partitions {λ} = {λ_{1}, λ_{2}, λ_{3}}, {μ} = {μ_{1}, μ_{2}} and {ν} = {ν_{1}, ⋯, ν_{5}} satisfy the following conditions:

\{ \begin{matrix}ν_{4} ≤ λ_{3} ≤ λ_{2} ≤ ν_{3} ≤ ν_{2} ≤ λ_{1} ≤ ν_{1}, \\ λ_{2} + μ_{1} + 1 ≤ ν_{2} + ν_{4}, \\ λ_{3} + μ_{2} ≤ ν_{3} + ν_{4} ≤ min \{λ_{2} + μ_{2}, λ_{3} + μ_{1}\}, \\ ν_{2} + ν_{5} ≤ min \{λ_{2} + μ_{2}, λ_{3} + μ_{1}\}, \\ λ_{3} + ν_{1} ≤ λ_{1} + λ_{2}. \end{matrix}

Then after some calculations, we know that the Littlewood-Richardson polynomial for this case is given by

\frac{1}{2} \{(−2λ_{1} − λ_{3} − μ_{1} + 2ν_{1} + ν_{2} + ν_{5})N + 2\}\{(λ_{3} + μ_{1} − ν_{2} − ν_{5})N + 1\}

+ (λ_{1} + λ_{2} − λ_{3} − ν_{1})N{(ν_{1} − λ_{1})N + 1}

+ \frac{1}{2} (λ_{3} + μ_{1} − ν_{3} − ν_{4})N\{(−2λ_{1} − λ_{3} − μ_{1} + 2ν_{1} + ν_{3} + ν_{4})N + 1\}.

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