A combinatorial rule for the schur function expansion of the PlethysmS(1^a,b)[P_k]

Quasisymmetric expansions of Schur-function plethysms

“…There is a purely combinatorial way to convert the Q-expansion of any symmetric function to a Schur expansion [8]. We illustrate this technique here by determining the coefficient of s (4,3,2) in the plethysm s (1,1,1) [s (2,1) ]. We begin by recalling the necessary details from [8].…”

“…Then by Theorems 11 and 15 of [8], x λ = α K * n (α, λ)y α . As illustrated in Figure 3, there are four flat special rim-hook tableaux of shape (4,3,2). Taking into account their signs, we therefore have (6.1) x 432 = y 432 − y 414 − y 252 + y 216 .…”

“…Numerous algorithms have been developed for computing the Schur expansion of a plethysm (most recently [5,6,25]), and several specific cases and extremal results have been obtained [2,3,4,15,22,26]. But an explicit combinatorial formula for the full Schur expansion of an arbitrary plethysm s λ [s μ ] has proved elusive.…”

“…For λ =(5,5,3,2) and α = (4, 2, 1, 8), we have K * 15 (α, λ) = −1. The relevant flat special rim-hook tableau is shown inFigure 2.…”

Shifted Set Families, Degree Sequences, and Plethysm

“…The following swinging operations on K result in a different shifted family. First swing 145 with respect to (2,4) which replaces 145 with 125. Next swing 156 with respect to (3,5) which replaces 156 with 136.…”