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This paper constructs two involutions on tip-augmented plane trees, as defined by Donaghey, that interchange two distinct types of leaves while preserving all other leaves. These two involutions provide bijective explanations addressing a…
Let $(\mathsf{baxt}_n,~^\sharp)$ be the Baxter monoid of finite rank $n$ with Sch\"{u}tzenberger's involution $^{\sharp}$. In this paper, it is shown that $(\mathsf{baxt}_n,~^\sharp)$ admits a faithful representation by an involution monoid…
We define a bijection that transforms an alternating sign matrix A with one -1 into a pair (N,E) where N is a (so called) ``neutral'' alternating sign matrix (with one -1) and E is an integer. The bijection preserves the classical…
We study the Schur polynomial expansion of a family of symmetric polynomials related to the refined enumeration of alternating sign matrices with respect to their inversion number, complementary inversion number and the position of the…
We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.
The spaces of triangulations of a given manifold have been widely studied. The celebrated theorem of Pachner~\cite{Pachner} says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves, or Pachner…
Topological protection of quantum correlations opens new horizons and opportunities in quantum technologies. A variety of topological effects has recently been observed in qubit networks. However, the experimental identification of the…
We study certain bijection between plane partitions and $\mathbb{N}$-matrices. As applications, we prove a Cauchy-type identity for generalized dual Grothendieck polynomials. We introduce two statistics on plane partitions, whose generating…
We develop a new bijective framework for the enumeration of bipartite planar maps with control on the degree distribution of black and white vertices. Our approach builds on the blossoming-tree paradigm, introducing a family of orientations…
In this paper, we construct a bijection from a set of bounded free Motzkin paths to a set of bounded Motzkin prefixes that induces a bijection from a set of bounded free Dyck paths to a set of bounded Dyck prefixes. We also give bijections…
The variety of complete quadrics is the wonderful compactification of $GL_n/O_n$ and admits a cell decomposition into Borel orbits indexed by combinatorial objects called $\mu$-involutions. We study Coxeter-theoretic properties of…
Following the method of combinatorial telescoping for alternating sums given by Chen, Hou and Mu, we present a combinatorial telescoping approach to partition identities on sums of positive terms. By giving a classification of the…
In this short note, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by L.…
We settle a question of Bressoud concerning the existence of an explicit bijection from a class of oriented square-ice graphs to a class of tournaments. We give an algorithm constructing such a bijection.
Set partitions avoiding $k$-crossing and $k$-nesting have been extensively studied from the aspects of both combinatorics and mathematical biology. By using the generating tree technique, the obstinate kernel method and Zeilberger's…
This paper is the second in a series of planned papers which provide first bijective proofs of alternating sign matrix results. Based on the main result from the first paper, we construct a bijective proof of the enumeration formula for…
We propose a new more efficient method for the computation of two-sided Gr\"obner bases of ideals and bimodules shifting the problem to the enveloping algebra. Arising from the ideas this method involves, we introduce the notion of…
This paper describes a Buchberger-style algorithm to compute a Groebner basis of a polynomial ideal, allowing for a selection strategy based on "signatures". We explain how three recent algorithms can be viewed as different strategies for…
This paper highlights three known identities, each of which involves sums over alternating sign matrices. While proofs of all three are known, the only known derivations are as corollaries of difficult results. The simplicity and natural…
We consider a specific %piecewise rotation of the plane that is continuous on two half-planes, class of piecewise rotations of the plane that are continuous on two half-planes, as studied in \cite{Bosh.Goet.03}, \cite{Goet.Quas.09} and…