Partition division maps, symmetric functions and positivity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914586731479040 |
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| author | Alexandersson, Per Dai, Lilan |
| author_facet | Alexandersson, Per Dai, Lilan |
| contents | We study a linear map on symmetric functions that ``divides'' a partition by a positive integer $k$, sending a Schur function indexed by a partition of $kn$ to a symmetric function indexed by partitions of $n$. We determine its Schur expansion explicitly for Schur and skew Schur functions, showing that the coefficients are enumerated by a new family of combinatorial objects, called $k$-Yamanouchi tableaux, which generalize the classical ballot (Yamanouchi) tableaux appearing in the Littlewood--Richardson rule.
We also study the images of elementary symmetric functions under this map, derive the power-sum expansion of their $ω$-images, and establish power-sum positivity. A further application establishes a connection to work of Tewodros Amdeberhan, John Shareshian, and Richard Stanley on alternating permutations and Euler numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25440 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Partition division maps, symmetric functions and positivity Alexandersson, Per Dai, Lilan Combinatorics Representation Theory 05E05, 05A17, 05A19 We study a linear map on symmetric functions that ``divides'' a partition by a positive integer $k$, sending a Schur function indexed by a partition of $kn$ to a symmetric function indexed by partitions of $n$. We determine its Schur expansion explicitly for Schur and skew Schur functions, showing that the coefficients are enumerated by a new family of combinatorial objects, called $k$-Yamanouchi tableaux, which generalize the classical ballot (Yamanouchi) tableaux appearing in the Littlewood--Richardson rule. We also study the images of elementary symmetric functions under this map, derive the power-sum expansion of their $ω$-images, and establish power-sum positivity. A further application establishes a connection to work of Tewodros Amdeberhan, John Shareshian, and Richard Stanley on alternating permutations and Euler numbers. |
| title | Partition division maps, symmetric functions and positivity |
| topic | Combinatorics Representation Theory 05E05, 05A17, 05A19 |
| url | https://arxiv.org/abs/2604.25440 |