We introduce a new family of Schur positive symmetric functions, which are defined as sums over totally symmetric plane partitions. In the first part, we show that, for , this family is equal to a multivariate generating function involving variables of objects that extend alternating sign matrices (ASMs), which have recently been introduced by the authors. This establishes a new connection between ASMs and a class of plane partitions, thereby complementing the fact that ASMs are equinumerous with totally symmetric self-complementary plane partitions as well as with descending plane partitions. The proof is based on a new antisymmetrizer-to-determinant formula for which we also provide a combinatorial proof, and, although this proof is complicated, it is an important step forward as it is very hard to find combinatorial proofs in this field. In the second part, we relate three specialisations of to weighted enumerations of certain well-known classes of column strict shifted plane partitions that generalise descending plane partitions.
Revised:
Accepted:
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Keywords: alternating sign matrices, column strict shifted plane partitions, totally symmetric plane partitions, Schur polynomials, Catalan numbers
Fischer, Ilse 1; Schreier-Aigner, Florian 1
CC-BY 4.0
@article{ALCO_2024__7_5_1319_0,
author = {Fischer, Ilse and Schreier-Aigner, Florian},
title = {Alternating sign matrices and totally symmetric plane partitions},
journal = {Algebraic Combinatorics},
pages = {1319--1345},
publisher = {The Combinatorics Consortium},
volume = {7},
number = {5},
year = {2024},
doi = {10.5802/alco.374},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.374/}
}
TY - JOUR AU - Fischer, Ilse AU - Schreier-Aigner, Florian TI - Alternating sign matrices and totally symmetric plane partitions JO - Algebraic Combinatorics PY - 2024 SP - 1319 EP - 1345 VL - 7 IS - 5 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.374/ DO - 10.5802/alco.374 LA - en ID - ALCO_2024__7_5_1319_0 ER -
%0 Journal Article %A Fischer, Ilse %A Schreier-Aigner, Florian %T Alternating sign matrices and totally symmetric plane partitions %J Algebraic Combinatorics %D 2024 %P 1319-1345 %V 7 %N 5 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.374/ %R 10.5802/alco.374 %G en %F ALCO_2024__7_5_1319_0
Fischer, Ilse; Schreier-Aigner, Florian. Alternating sign matrices and totally symmetric plane partitions. Algebraic Combinatorics, Volume 7 (2024) no. 5, pp. 1319-1345. doi : 10.5802/alco.374. https://alco.centre-mersenne.org/articles/10.5802/alco.374/
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