Difference operators for partitions under the Littlewood decomposition

Abstract

Abstract. The concept of t-difference operator for functions of partitions is introduced to prove a generalization of Stanley's theorem on polynomiality of Plancherel averages of symmetric functions related to contents and hook lengths. Our extension uses a generalization of the notion of Plancherel measure, based on walks in the Young lattice with steps given by the addition of t-hooks. It is well-known that the hook lengths of multiples of t can be characterized by the Littlewood decomposition. Our study gives some further information on the contents and hook lengths of other congruence classes modulo t.

New hook-content formulas for strict partitions

“…The first author conjectured [4] that P (n) = 1 n! |ν|=n f 2 is always a polynomial in n for all k ∈ N, which was generalized and proved by Stanley [16], and later generalized in [7] (see also [2,5,6,8,10,11,12,13]).…”

Polynomiality of Certain Average Weights for Oscillating Tableaux

“…For each box in the Young diagram of the partition λ, let h and c be its hook length and content respectively (see [18,26]). In a preparing paper, by applying results from the study of difference operators on functions of partitions [7,11,12], we will establish the following two explicit formulas with very complicated proofs for the average weights related to hook lengths and contents: (c 2 − j 2 ) = (2r)! (r + 1)!…”

Polynomiality of Plancherel averages of hook-content summations for strict, doubled distinct and self-conjugate partitions

“…To prove Lemma 4.4, we recall some results on the multisets of hook lengths and contents, obtained in [3]. Suppose that a given t-core partition µ has 01-sequence w(µ) = (a µ,j ) j∈Z .…”

“…Suppose that a given t-core partition µ has 01-sequence w(µ) = (a µ,j ) j∈Z . For 0 ≤ i ≤ t − 1 we define [3] b i := b i (µ) = min{j ∈ Z : j ≡ i(mod t), a µ,j = 1}. ).…”

“…Lemma 4.6 (Lemma 5.4 of [3]). Let 0 ≤ i ≤ t−1, λ and λ + be two usual partitions whose images of the Littlewood decomposition are (λ t-core ; λ 0 , λ 1 , .…”