ALGEBRAIC COMBINATORICS

no. 1
Combinatorics of the Delta conjecture at q=-1
Corteel, Sylvie1; Josuat-Vergès, Matthieu1; Vanden Wyngaerd, Anna2
1 IRIF Université Paris Cité 8 Place Aurélie Nemours 75205 Paris Cedex 13 (France)
2 Département de Mathématiques Université Libre de Bruxelles Campus de la Plaine Boulevard du Triomphe 1050 Ixelles (Belgique)
Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 17-35.

In the context of the shuffle theorem, many classical integer sequences appear with a natural refinement by two statistics q and t: for example the Catalan and Schröder numbers. In particular, the bi-graded Hilbert series of diagonal harmonics is a q,t-analog of (n+1) n-1 (and can be written in terms of symmetric functions via the nabla operator). The motivation for this work is the observation that at q=-1, this q,t-analog becomes a t-analog of Euler numbers, a famous integer sequence that counts alternating permutations. We prove this observation via a more general statement, that involves the Delta operator on symmetric functions (on one side), and new combinatorial statistics on permutations involving peaks and valleys (on the other side). An important tool are the schedule numbers of a parking function first introduced by Hicks; and expanded upon by Haglund and Sergel. Other empirical observations suggests that non negativity at q=-1 holds in far greater generality.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.329
Classification: 05A05, 05A30, 05E05
Keywords: Delta conjecture, Macdonald polynomials, permutations, schedule numbers
Corteel, Sylvie 1; Josuat-Vergès, Matthieu 1; Vanden Wyngaerd, Anna 2

1 IRIF Université Paris Cité 8 Place Aurélie Nemours 75205 Paris Cedex 13 (France)
2 Département de Mathématiques Université Libre de Bruxelles Campus de la Plaine Boulevard du Triomphe 1050 Ixelles (Belgique)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Corteel, Sylvie; Josuat-Vergès, Matthieu; Vanden Wyngaerd, Anna. Combinatorics of the Delta  conjecture at $q=-1$. Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 17-35. doi : 10.5802/alco.329. https://alco.centre-mersenne.org/articles/10.5802/alco.329/

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