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Bliem and Kousidis (arXiv:1109.4624) recently considered a family of random variables whose distributions are given by the generalized Galois numbers (after normalization). We give probabilistic interpretations of these random variables,…
We extend the polynomial approach to hook length formula proposed in a recent joint paper with K\'arolyi, Nagy and Volkov to several other problems of the same type, including number of paths formula in the Young graph of strict partitions.
Using the symmetry reductions of the self-dual Yang-Mills (SDYM) equations in (2+2) dimensions, we introduce new integrable equations which are nonautonomous versions of the chiral model in (2+1) dimensions, generalized nonlinear…
We prove a constant term theorem which is useful for finding weight polynomials for Ballot/Motzkin paths in a strip with a fixed number of arbitrary `decorated' weights as well as an arbitrary `background' weight. Our CT theorem, like…
The $m$-Tamari lattices $\mathcal{T}_{n}^{(m)}$, introduced by Bergeron and Pr{\'e}ville-Ratelle, are defined as a poset of $m$-Dyck paths equipped with the generalized rotation order, and constitute a Fuss-Catalan generalization of the…
We give formulas for enumerating directed paths in the graded poset of semi-magic squares of size three. We give two applications of these formulas: an advanced example of Vandermonde convolution for finite graded posets, and a direct…
A combinatorial theory for type $R_I$ orthogonal polynomials is given. The ingredients include weighted generalized Motzkin paths, moments, continued fractions, determinants, and histories. Several explicit examples in the Askey scheme are…
Ilse Fischer and the second author introduced in [Algebr. Comb. 7 (2024), no. 5, 1319-1345] a two parameter family of polynomials defined as sums over totally symmetric plane partitions and connected to alternating sign matrices and…
We introduce a graph structure on Euclidean polytopes. The vertices of this graph are the $d$-dimensional polytopes contained in $\mathbb{R}^d$ and its edges connect any two polytopes that can be obtained from one another by either…
The number of down-steps between pairs of up-steps in $k_t$-Dyck paths, a generalization of Dyck paths consisting of steps $\{(1, k), (1, -1)\}$ such that the path stays (weakly) above the line $y=-t$, is studied. Results are proved…
In a recent paper published in Nature, Y.I. Sobolev et al. introduced the concept of trajectoids: convex, rigid objects, which roll without slip or spin on a flat plane along a prescribed periodic, unbounded planar path. A geometric…
We introduce the minor-closed, dual-closed class of multi-path matroids. We give a polynomial-time algorithm for computing the Tutte polynomial of a multi-path matroid, we describe their basis activities, and we prove some basic structural…
We study two different one-parameter generalizations of Littlewood--Richardson coefficients, namely Hall polynomials and generalized inverse Kostka polynomials, and derive new combinatorial formulae for them. Our combinatorial expressions…
Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They were originally defined only in odd characteristic, but recently Zhou introduced a definition in even characteristic which…
We prove that on the set of lattice paths with steps N=(0,1) and E=(1,0) that lie between two fixed boundaries T and B (which are themselves lattice paths), the statistics `number of E steps shared with B' and `number of E steps shared with…
Koutschan, Krattenthaler and Schlosser recently considered a family of binomial determinants. In this work, we give combinatorial interpretations of two subclasses of these determinants in terms of domino tilings and nonintersecting lattice…
We show that recent determinant evaluations involving Catalan numbers and generalisations thereof have most convenient explanations by combining the Lindstr\"om-Gessel-Viennot theorem on non-intersecting lattice paths with a simple…
We discuss the combinatorics of the decorated Dyck paths appearing in the Delta conjecture framework of Haglund, Remmel and Wilson (2015) and Zabrocki (2016), by introducing two new statistics, bounce and bounce'. We then provide plethystic…
In [J. Phys. A: Math. Theor. 45 (2012)], while looking for spin chains that admit perfect state transfer, Vinet and Zhedanov found an apparently new sequence of orthogonal polynomials, that they called para-Krawtchouk polynomials, defined…
We introduce two classes of discrete polynomials and construct discrete equations admitting a Lax representation in terms of these polynomials. Also we give an approach which allows to construct lattice integrable hierarchies in its…