FURTHER ENUMERATION RESULTS CONCERNING A RECENT EQUIVALENCE OF RESTRICTED INVERSION SEQUENCES

Toufik Mansour (1) , Mark Shattuck

1 University of Haifa [Haifa]

Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class In(≥, =, >) is the same as that of n − 1 − asc on the class In(>, =, ≥). In this paper, we consider some further enumerative aspects related to this equivalence. In particular, we find recurrence relations for the joint distribution on In(≥, =, >) of asc and desc along with two other parameters, and do the same for n − 1 − asc and desc on In(>, =, ≥). By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and provides a new proof of) the recent result |In(≥, =, >)| = |In(>, =, ≥)|. In both cases, an algorithm is formulated for computing the generating function of the asc distribution on members of each respective class having a fixed number of descents.

Mots clés: July 22; 2021. 2010 Mathematics Subject Classification. 05A15; 05A05; 65Q20; 65Q30 pattern avoidance; combinatorial statistic; kernel method; inversion sequence

Domaines: Combinatoire [math.CO]; Physique mathématique [math-ph]

Toufik Mansour, Mark Shattuck. FURTHER ENUMERATION RESULTS CONCERNING A RECENT EQUIVALENCE OF RESTRICTED INVERSION SEQUENCES. 2021. ⟨hal-03295362v1⟩