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 -