5 research outputs found
The Binomial-Stirling-Eulerian Polynomials
We introduce the binomial-Stirling-Eulerian polynomials, denoted
, which encompass binomial coefficients, Eulerian
numbers and two Stirling statistics: the left-to-right minima and the
right-to-left minima. When , these polynomials reduce to the
binomial-Eulerian polynomials , originally named by
Shareshian and Wachs and explored by Chung-Graham-Knuth and
Postnikov-Reiner-Williams. We investigate the -positivity of
from two aspects: firstly by employing the
grammatical calculus introduced by Chen; and secondly by constructing a new
group action on permutations. These results extend the symmetric Eulerian
identity found by Chung, Graham and Knuth, and the -positivity of
first demonstrated by Postnikov, Reiner and Williams.Comment: 18 page. Any comments are welcome. arXiv admin note: text overlap
with arXiv:2310.0105