Set partitions, tableaux, and subspace profiles under regular diagonal matrices
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2021
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866929500436037632 |
|---|---|
| author | Prasad, Amritanshu Ram, Samrith |
| author_facet | Prasad, Amritanshu Ram, Samrith |
| contents | We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. This enumeration formula is a combinatorial solution to a problem introduced by Bender, Coley, Robbins and Rumsey. At $1$, they count set partitions with specified block sizes. At $0$, they count standard tableaux of specified shape. At $-1$, they count standard shifted tableaux of a specified shape. These polynomials are generated by a new statistic on set partitions (called the interlacing number) as well as a polynomial statistic on standard tableaux. They allow us to express $q$-Stirling numbers of the second kind as sums over standard tableaux and as sums over set partitions.
For partitions whose parts are at most two, these polynomials are the non-zero entries of the Catalan triangle associated to the $q$-Hermite orthogonal polynomial sequence. In particular, when all parts are equal to two, they coincide with the polynomials defined by Touchard that enumerate chord diagrams by the number of crossings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_00479 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Set partitions, tableaux, and subspace profiles under regular diagonal matrices Prasad, Amritanshu Ram, Samrith Combinatorics 05A15, 05A18, 05A30, 15B33, 33C45 We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. This enumeration formula is a combinatorial solution to a problem introduced by Bender, Coley, Robbins and Rumsey. At $1$, they count set partitions with specified block sizes. At $0$, they count standard tableaux of specified shape. At $-1$, they count standard shifted tableaux of a specified shape. These polynomials are generated by a new statistic on set partitions (called the interlacing number) as well as a polynomial statistic on standard tableaux. They allow us to express $q$-Stirling numbers of the second kind as sums over standard tableaux and as sums over set partitions. For partitions whose parts are at most two, these polynomials are the non-zero entries of the Catalan triangle associated to the $q$-Hermite orthogonal polynomial sequence. In particular, when all parts are equal to two, they coincide with the polynomials defined by Touchard that enumerate chord diagrams by the number of crossings. |
| title | Set partitions, tableaux, and subspace profiles under regular diagonal matrices |
| topic | Combinatorics 05A15, 05A18, 05A30, 15B33, 33C45 |
| url | https://arxiv.org/abs/2112.00479 |