Sects and Lattice Paths over the Lagrangian Grassmannian

We examine Borel subgroup orbits in the classical symmetric space of type $CI$, which are parametrized by skew symmetric $(n, n)$-clans. We describe bijections between such clans, certain weighted lattice paths, and pattern-avoiding signed involutions, and we give a cell decomposition of the symmetric space in terms of collections of clans called sects. The largest sect with a conjectural closure order is isomorphic (as a poset) to the Bruhat order on partial involutions.

A Cornucopia of Quasi-Yamanouchi Tableaux

Quasi-Yamanouchi tableaux are a subset of semistandard Young tableaux and refine standard Young tableaux. They are closely tied to the descent set of standard Young tableaux and were introduced by Assaf and Searles to tighten Gessel's fundamental quasisymmetric expansion of Schur functions. The descent set and descent statistic of standard Young tableaux repeatedly prove themselves useful to consider, and as a result, quasi-Yamanouchi tableaux make appearances in many ways outside of their original purpose. Some examples, which we present in this paper, include the Schur expansion of Jack polynomials, the decomposition of Foulkes characters, and the bigraded Frobenius image of the coinvariant algebra. While it would be nice to have a product formula enumeration of quasi-Yamanouchi tableaux in the way that semistandard and standard Young tableaux do, it has previously been shown by the author that there is little hope on that front. The goal of this paper is to address a handful of the numerous alternative enumerative approaches. In particular, we present enumerations of quasi-Yamanouchi tableaux using $q$-hit numbers, semistandard Young tableaux, weighted lattice paths, and symmetric polynomials, as well as the fundamental quasisymmetric and monomial quasisymmetric expansions of their Schur generating function.

A Generalization of the $k$-Bonacci Sequence from Riordan Arrays

In this article, we introduce a family of weighted lattice paths, whose step set is $\{H=(1,0), V=(0,1), D_1=(1,1), \dots, D_{m-1}=(1,m-1)\}$. Using these lattice paths, we define a family of Riordan arrays whose sum on the rising diagonal is the $k$-bonacci sequence. This construction generalizes the Pascal and Delannoy Riordan arrays, whose sum on the rising diagonal is the Fibonacci and tribonacci sequence, respectively. From this family of Riordan arrays we introduce a generalized $k$-bonacci polynomial sequence, and we give a lattice path combinatorial interpretation of these polynomials. In particular, we find a combinatorial interpretation of tribonacci and tribonacci-Lucas polynomials.

Lattice Paths, Sampling Without Replacement, and Limiting Distributions

In this work we consider weighted lattice paths in the quarter plane ${\Bbb N}_0\times{\Bbb N}_0$. The steps are given by $(m,n)\to(m-1,n)$, $(m,n)\to(m,n-1)$ and are weighted as follows: $(m,n)\to(m-1,n)$ by $m/(m+n)$ and step $(m,n)\to(m,n-1)$ by $n/(m+n)$. The considered lattice paths are absorbed at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$. We provide explicit formulæ for the sum of the weights of paths, starting at $(m,n)$, which are absorbed at a certain height $k$ at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$, using a generating functions approach. Furthermore these weighted lattice paths can be interpreted as probability distributions arising in the context of Pólya-Eggenberger urn models, more precisely, the lattice paths are sample paths of the well known sampling without replacement urn. We provide limiting distribution results for the underlying random variable, obtaining a total of five phase changes.