Scrambled Vandermonde convolutions of Gaussian polynomials
(2022) In Discrete Mathematics 345(12).- Abstract
It is well known that Gaussian polynomials (i.e., q-binomials) describe the distribution of the [Formula presented] statistic on monotone paths in a rectangular grid. We introduce two new statistics, [Formula presented] and [Formula presented]; attach “ornaments” to the grid that scramble the values of [Formula presented] in specific fashion; and re-evaluate these statistics, in order to argue that all scrambled versions of the [Formula presented] statistic are equidistributed with [Formula presented]. Our main result is a representation of the generating function for the bi-statistic [Formula presented] as a new, two-variable Vandermonde convolution of the original Gaussian polynomial. The proof relies on explicit bijections between... (More)
It is well known that Gaussian polynomials (i.e., q-binomials) describe the distribution of the [Formula presented] statistic on monotone paths in a rectangular grid. We introduce two new statistics, [Formula presented] and [Formula presented]; attach “ornaments” to the grid that scramble the values of [Formula presented] in specific fashion; and re-evaluate these statistics, in order to argue that all scrambled versions of the [Formula presented] statistic are equidistributed with [Formula presented]. Our main result is a representation of the generating function for the bi-statistic [Formula presented] as a new, two-variable Vandermonde convolution of the original Gaussian polynomial. The proof relies on explicit bijections between differently ornated paths.
(Less)
- author
- Aspenberg, Magnus LU and Pérez, Rodrigo A.
- organization
- publishing date
- 2022
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Gaussian polynomials, Integer partitions, Lattice paths, q-binomials, q-Vandermonde convolution
- in
- Discrete Mathematics
- volume
- 345
- issue
- 12
- article number
- 113064
- publisher
- Elsevier
- external identifiers
-
- scopus:85134416139
- ISSN
- 0012-365X
- DOI
- 10.1016/j.disc.2022.113064
- language
- English
- LU publication?
- yes
- id
- dfcfcbff-6b8c-4458-9ada-e326cb256d60
- date added to LUP
- 2022-08-25 15:50:43
- date last changed
- 2022-08-25 15:50:43
@article{dfcfcbff-6b8c-4458-9ada-e326cb256d60,
abstract = {{<p>It is well known that Gaussian polynomials (i.e., q-binomials) describe the distribution of the [Formula presented] statistic on monotone paths in a rectangular grid. We introduce two new statistics, [Formula presented] and [Formula presented]; attach “ornaments” to the grid that scramble the values of [Formula presented] in specific fashion; and re-evaluate these statistics, in order to argue that all scrambled versions of the [Formula presented] statistic are equidistributed with [Formula presented]. Our main result is a representation of the generating function for the bi-statistic [Formula presented] as a new, two-variable Vandermonde convolution of the original Gaussian polynomial. The proof relies on explicit bijections between differently ornated paths.</p>}},
author = {{Aspenberg, Magnus and Pérez, Rodrigo A.}},
issn = {{0012-365X}},
keywords = {{Gaussian polynomials; Integer partitions; Lattice paths; q-binomials; q-Vandermonde convolution}},
language = {{eng}},
number = {{12}},
publisher = {{Elsevier}},
series = {{Discrete Mathematics}},
title = {{Scrambled Vandermonde convolutions of Gaussian polynomials}},
url = {{http://dx.doi.org/10.1016/j.disc.2022.113064}},
doi = {{10.1016/j.disc.2022.113064}},
volume = {{345}},
year = {{2022}},
}