Related papers: Overlapping Pfaffians

Let $\overline{\mathrm{spt}}k(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part, say $s(\pi)$, appears $k$ times and every overlined part is bigger than $s(\pi)$. Let $\overline{\mathrm{spt}}k_o(n)$ denote…

In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…

Cauchy's interlace theorem states that the characteristic polynomial of a symmetric matrix is interlaced by the characteristic polynomial of any principle submatrix. We prove this in two sentences using only the linearity of the…

We prove that a curious generating series identity implies Faber's intersection number conjecture (by showing that it implies a combinatorial identity already given in arXiv:1902.02742) and give a new proof of Faber's conjecture by directly…

This is a study on pattern Hopf algebras in combinatorial structures. We introduce the notion of combinatorial presheaf, by adapting the algebraic framework of species to the study of substructures in combinatorics. Afterwards, we consider…

We formalize algorithms computing Pfaffian in the theory of bounded arithmetic for sharpL which is based on Berkowitz algorithm for the determinant. We also prove relations among Pfaffian properties. Furthermore, we give an algorithm for…

We introduce and study a cyclically invariant polynomial which is an analog of the classical tridiagonal determinant usually called the continuant. We prove that this polynomial can be calculated as the Pfaffian of a skew-symmetric matrix.…

An alternating sign matrix is a square matrix with entries 1, 0 and -1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the…

It is well known that every smooth cubic threefold is the zero locus of the Pfaffian of a 6 x 6 skew-symmetric matrix of linear forms in P^4. To compactify the space of such Pfaffian representations of a given cubic and to study the…

The rank 4 locus of a general skew-symmetric 7x7 matrix gives the pfaffian variety in P^20 which is not defined as a complete intersection. Intersecting this with a general P^6 gives a Calabi-Yau manifold. An orbifold construction seems to…

We present a proof of a combinatorial conjecture from the second author's Ph.D. thesis. The proof relies on binomial and multinomial sums identities. We also discuss the relevance of the conjecture in the context of PAC-Bayesian machine…

A Fibonacci pair $F_s(w,x)$ of rank $s$ is a pair $s \times s$ nonsingular matrices such that $wx=xw$ and that the entries of $aw^n$ and $axw^m$ are polynomials of Fibonacci or Lucas numbers for some nonzero $a$. We construct identities…

The purpose of this note is to exhibit clearly how the "graphical condensation" identities of Kuo, Yan, Yeh and Zhang follow from classical Pfaffian identities by the Kasteleyn-Percus method for the enumeration of matchings. Knuth termed…

In this paper, we introduce the concept of the over-Mahonian number, which counts the overlined permutations of length $n$ with $k$ inversions, allowing the first elements associated with the inversions to be independently overlined or not.…

It is known that there exist hyperplane arrangements with same underlying matroid that admit non-homotopy equivalent complement manifolds. In this work we show that, in any rank, complex central hyperplane arrangements with up to 7…

Some skew-symmetrizable integer exchange matrices are associated to ideal (tagged) triangulations of marked bordered surfaces. These exchange matrices admits unfoldings to skew-symmetric matrices. We develop an combinatorial algorithm that…

Two matrices are said non-overlapping if one of them can not be put on the other one in a way such that the corresponding entries coincide. We provide a set of non-overlapping binary matrices and a formula to enumerate it which involves the…

Equifacetal simplices, all of whose codimension one faces are congruent to one another, are studied. It is shown that the isometry group of such a simplex acts transitively on its set of vertices, and, as an application, equifacetal…

Using a probabilistic approach, we derive some interesting combinatorial identities involving gamma and beta functions. These results generalize certain well-known combinatorial identities involving binomial coefficients and special…

An identity that is reminiscent of the Littlewood identity plays a fundamental role in recent proofs of the facts that alternating sign triangles are equinumerous with totally symmetric self-complementary plane partitions and that…