The dual of number sequences, Riordan polynomials, and Sheffer polynomials

In this paper we introduce different families of numerical and polynomial sequences by using Riordan pseudo involutions and Sheffer polynomial sequences. Many examples are given including dual of Hermite numbers and polynomials, dual of Bell numbers and polynomials, among other. The coefficients of some of these polynomials are related to the counting of different families of set partitions and permutations. We also studied the dual of Catalan numbers and dual of Fuss-Catalan numbers, giving several combinatorial identities.

Generalized Catalan Numbers from Hypergraphs

The Catalan numbers $C_{n} \in \{1,1,2,5,14,42,\dots \}$ form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting rooted plane trees and noncrossing set partitions. They also arise in the GUE matrix model as the leading coefficient of certain polynomials, a connection closely related to the plane trees and noncrossing set partitions interpretations. In this paper we define a generalization of the Catalan numbers. In fact we actually define an infinite collection of generalizations $C_{n}^{(m)}$, $m\geq 1$, with $C_{n}^{(1)}$ equal to the usual Catalans $C_{n}$; the sequence $C_{n}^{(m)}$ comes from studying certain matrix models attached to hypergraphs. We also give some combinatorial interpretations of these numbers.

Quarter-BPS states, multi-symmetric functions and set partitions

We give a construction of general holomorphic quarter BPS operators in $$ \mathcal{N} $$ N = 4 SYM at weak coupling with U(N) gauge group at finite N. The construction employs the Möbius inversion formula for set partitions, applied to multi-symmetric functions, alongside computations in the group algebras of symmetric groups. We present a computational algorithm which produces an orthogonal basis for the physical inner product on the space of holomorphic operators. The basis is labelled by a U(2) Young diagram, a U(N) Young diagram and an additional plethystic multiplicity label. We describe precision counting results of quarter BPS states which are expected to be reproducible from dual computations with giant gravitons in the bulk, including a symmetry relating sphere and AdS giants within the quarter BPS sector. In the case n ≤ N (n being the dimension of the composite operator) the construction is analytic, using multi-symmetric functions and U(2) Clebsch-Gordan coefficients. Counting and correlators of the BPS operators can be encoded in a two-dimensional topological field theory based on permutation algebras and equipped with appropriate defects.

Harmonic Bases for Generalized Coinvariant Algebras

Let $k \leq n$ be nonnegative integers and let $\lambda$ be a partition of $k$. S. Griffin recently introduced a quotient $R_{n,\lambda}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which simultaneously generalizes the Delta Conjecture coinvariant rings of Haglund-Rhoades-Shimozono and the cohomology rings of Springer fibers studied by Tanisaki and Garsia-Procesi. We describe the space $V_{n,\lambda}$ of harmonics attached to $R_{n,\lambda}$ and produce a harmonic basis of $R_{n,\lambda}$ indexed by certain ordered set partitions $\mathcal{OP}_{n,\lambda}$. The combinatorics of this basis is governed by a new extension of the Lehmer code of a permutation to $\mathcal{OP}_{n, \lambda}$.

Sorting with Pattern-Avoiding Stacks: The 132-Machine

This paper continues the analysis of the pattern-avoiding sorting machines recently introduced by Cerbai, Claesson and Ferrari (2020). These devices consist of two stacks, through which a permutation is passed in order to sort it, where the content of each stack must at all times avoid a certain pattern. Here we characterize and enumerate the set of permutations that can be sorted when the first stack is $132$-avoiding, solving one of the open problems proposed by the above mentioned authors. To that end we present several connections with other well known combinatorial objects, such as lattice paths and restricted growth functions (which encode set partitions). We also provide new proofs for the enumeration of some sets of pattern-avoiding restricted growth functions and we expect that the tools introduced can be fruitfully employed to get further similar results.