TY - JOUR

T1 - Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers

AU - Aval, Jean-Christophe

AU - D'Adderio, Michele

AU - Dukes, Mark

AU - Hicks, Angela

AU - Le Borgne, Yvan

PY - 2014/4

Y1 - 2014/4

N2 - We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area, bounce) and (area, dinv) give rise to the same q,t-analogue of Narayana numbers which was introduced by two of the authors in [arXiv:1208.0024]. We prove the main conjectures of that paper: the q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q,t-Narayana polynomials which relates them to the famous diagonal harmonics.

AB - We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area, bounce) and (area, dinv) give rise to the same q,t-analogue of Narayana numbers which was introduced by two of the authors in [arXiv:1208.0024]. We prove the main conjectures of that paper: the q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q,t-Narayana polynomials which relates them to the famous diagonal harmonics.

KW - q,tq,t-Narayana

KW - parallelogram polyominoes

KW - parking functions

UR - http://arxiv.org/abs/1301.4803

UR - http://www.sciencedirect.com/science/article/pii/S0097316513001325

U2 - 10.1016/j.jcta.2013.09.001

DO - 10.1016/j.jcta.2013.09.001

M3 - Article

SN - 0097-3165

VL - 123

SP - 271

EP - 286

JO - Journal of Combinatorial Theory Series A

JF - Journal of Combinatorial Theory Series A

IS - 1

ER -