Related papers: Hook-valued tableau uncrowding and tableau switchi…
This paper investigates the independence polynomials arising from iterated strong products of cycle graphs, examining their algebraic symmetries and combinatorial structures. Leveraging modular arithmetic and Galois theory, we establish…
We study the Grothendieck group of the variety $X_{n,k}$ of spanning line configurations introduced by Pawlowski--Rhoades [arXiv:1711.08301] as a geometric model for the generalized coinvariant algebra $R_{n,k}$. Our first result is a…
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…
Buch, Kresch, Shimozono, Tamvakis, and Yong defined Hecke insertion to formulate a combinatorial rule for the expansion of the stable Grothendieck polynomials $G_\pi$ indexed by permutations in the basis of stable Grothendieck polynomials…
We discuss the combinatorics of decorated Dyck paths and decorated parallelogram polyominoes, extending to the decorated case the main results of both [Haglund 2004] and [Aval et al. 2014]. This settles in particular the cases…
Grothendieck polynomials, introduced by Lascoux and Sch\"utzenberger, are certain $K$-theory representatives for Schubert varieties. Symplectic Grothendieck polynomials, described more recently by Wyser and Yong, represent the $K$-theory…
A combinatorial description of the crystal B(infinity) for finite-dimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and…
Many automatic theorem provers are restricted to untyped logics, and existing translations from typed logics are bulky or unsound. Recent research proposes monotonicity as a means to remove some clutter when translating monomorphic to…
Patrias and Pylyavskyy introduced shifted Hecke insertion as an application of their theory of dual filtered graphs. We use shifted Hecke insertion to construct symmetric function representatives for the K-theory of the orthogonal…
Ge and Stefankovic have recently introduced a novel two-variable graph polynomial. When specialised to a bipartite graphs G and evaluated at the point (1/2,1) this polynomial gives the number of independent sets in the graph. Inspired by…
In the mid sixties, A. Grothendieck envisioned a vast generalization of Galois theory to systems of polynomials in several variables, motivic Galois theory, and introduced tannakian categories on this occasion. In characteristic zero,…
We confirm a conjecture of Gartland and Lokshtanov [arXiv:2007.08761]: if for a hereditary graph class $\mathcal{G}$ there exists a constant $k$ such that no member of $\mathcal{G}$ contains a $k$-creature as an induced subgraph or a…
Lascoux polynomials are $K$-theoretic analogues of the key polynomials. They both have combinatorial formulas involving tableaux: reverse set-valued tableaux ($\mathsf{RSVT}$) rule for Lascoux polynomials and reverse semistandard Young…
Let $G$ be an $n$-node graph without two disjoint odd cycles. The algorithm of Artmann, Weismantel and Zenklusen (STOC'17) for bimodular integer programs can be used to find a maximum weight stable set in $G$ in strongly polynomial time.…
We introduce a Grothendieck ring of higher Artin stacks generalizing the Grothendieck ring of algebraic varieties. We show that this ring is not trivial by noticing that it factors the invariant "number of rational points over a finite…
The symmetric Grothendieck polynomials generalize Schur polynomials and are Schur-positive by degree. Combinatorially this is manifested as the generalization of semistandard Young tableaux by set-valued tableaux. We define a (weak)…
Associated to any uniform finite layered graph Gamma there is a noncommutative graded quadratic algebra A(Gamma) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul.…
Recently, Diaconis, Ram and I created Markov chains out of the coproduct-then-product operator on combinatorial Hopf algebras. These chains model the breaking and recombining of combinatorial objects. Our motivating example was the…
The rook polynomial is a generating function that enumerates the number of ways to place rooks, with no two in the same row or column, on a collection of cells regarded as a pruned chessboard. In combinatorial commutative algebra, special…
This paper introduces a class of polynomial maps in Euclidean spaces, investigates the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets, studies the chaotic dynamical behavior and strange…