Let $r \ge 0$, and let $\lambda $ and $\mu $ be partitions such that $\lambda _1 \le r + 1$. We present a combinatorial interpretation of the plethysm coefficient $\langle s_\lambda , s_\mu [s_r] \rangle $. As a consequence, we solve the restriction problem for partitions with at most three columns. That is, for all partitions $\lambda $ with $\lambda _1 \le 3$, we find a combinatorial interpretation for the multiplicities of the irreducible $\mathfrak{S}_n$-submodules of the Schur module $\mathbb{S}^\lambda \mathbb{C}^n$, considered as an $\mathfrak{S}_n$-module.
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Keywords: restriction coefficients, restriction problem, symmetric functions, plethysm
CC-BY 4.0
Lee, Mitchell. Restriction coefficients for partitions with at most three columns. Algebraic Combinatorics, Volume 9 (2026) no. 2, pp. 543-555. doi: 10.5802/alco.476
@article{ALCO_2026__9_2_543_0,
author = {Lee, Mitchell},
title = {Restriction coefficients for partitions with at most three columns},
journal = {Algebraic Combinatorics},
pages = {543--555},
year = {2026},
publisher = {The Combinatorics Consortium},
volume = {9},
number = {2},
doi = {10.5802/alco.476},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.476/}
}
TY - JOUR AU - Lee, Mitchell TI - Restriction coefficients for partitions with at most three columns JO - Algebraic Combinatorics PY - 2026 SP - 543 EP - 555 VL - 9 IS - 2 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.476/ DO - 10.5802/alco.476 LA - en ID - ALCO_2026__9_2_543_0 ER -
%0 Journal Article %A Lee, Mitchell %T Restriction coefficients for partitions with at most three columns %J Algebraic Combinatorics %D 2026 %P 543-555 %V 9 %N 2 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.476/ %R 10.5802/alco.476 %G en %F ALCO_2026__9_2_543_0
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