14 research outputs found
Congruences for Taylor expansions of quantum modular forms
Recently, a beautiful paper of Andrews and Sellers has established linear
congruences for the Fishburn numbers modulo an infinite set of primes. Since
then, a number of authors have proven refined results, for example, extending
all of these congruences to arbitrary powers of the primes involved. Here, we
take a different perspective and explain the general theory of such congruences
in the context of an important class of quantum modular forms. As one example,
we obtain an infinite series of combinatorial sequences connected to the
"half-derivatives" of the Andrews-Gordon functions and with Kashaev's invariant
on torus knots, and we prove conditions under which the sequences
satisfy linear congruences modulo at least of primes of primes
Decomposing labeled interval orders as pairs of permutations
We introduce ballot matrices, a signed combinatorial structure whose
definition naturally follows from the generating function for labeled interval
orders. A sign reversing involution on ballot matrices is defined. We show that
matrices fixed under this involution are in bijection with labeled interval
orders and that they decompose to a pair consisting of a permutation and an
inversion table. To fully classify such pairs, results pertaining to the
enumeration of permutations having a given set of ascent bottoms are given.
This allows for a new formula for the number of labeled interval orders
On naturally labelled posets and permutations avoiding 12-34
A partial order on is naturally labelled (NL) if
implies . We establish a bijection between {3, 2+2}-free NL posets and
12-34-avoiding permutations, determine functional equations satisfied by their
generating function, and use series analysis to investigate their asymptotic
growth. We also exhibit bijections between 3-free NL posets and various other
objects, and determine their generating function. The connection between our
results and a hierarchy of combinatorial objects related to interval orders is
described.Comment: 19 page
A note on p-Ascent Sequences
Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes, and Kitaev in [1], who showed that ascent sequences of length n are in 1-to-1 correspondence with (2+2)-free posets of size n. In this paper, we introduce a generalization of ascent sequences, which we call p-ascent sequences, where p \geq 1. A sequence of non-negative integers is a p-ascent sequence if and for all , is at most p plus the number of ascents in . Thus, in our terminology, ascent sequences are 1-ascent sequences. We generalize a result of the authors in [9] by enumerating p-ascent sequences with respect to the number of 0s. We also generalize a result of Dukes, Kitaev, Remmel, and SteingrÃmsson in [4] by finding the generating function for the number of p-ascent sequences which have no consecutive repeated elements. Finally, we initiate the study of pattern-avoiding p-ascent sequences