Yan X Zhang

Adinkras for mathematicians

Adinkras are graphical tools created for the study of supersymmetry representations. Besides having in- herent interest for physicists, the study of adinkras has already shown connections with coding theory and Clifford algebras. Furthermore, adinkras offer many natural and accessible mathematical problems of combinatorial nature. We present the foundations for a mathematical audience, make new connections to other fields (homological algebra, poset theory, and polytopes), and solve some of these problems. Original results include the enumeration of all hyper- cube adinkras through dimension 5, the enumeration of odd dashings of adinkras for any dimension, and a connection between rankings and the chromatic polynomial for certain graphs. R´ esum´ e.

Matrices with restricted entries and

We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are

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-analogues of permutations

-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a

Enumeration of graded (3+1)-avoiding posets

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Separating hyperplanes of edge polytopes

-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank, and we frame some of our results in the context of Lie theory. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.

Adinkras for Mathematicians

Abstract The notion of ( 3 + 1 ) -avoidance has shown up in many places in enumerative combinatorics, but the natural goal of enumerating all ( 3 + 1 ) -avoiding posets remains open. In this paper, we enumerate graded ( 3 + 1 ) -avoiding posets for both reasonable definitions of the word “graded.” Our proof consists of a number of structural theorems followed by some generating function computations. We also provide asymptotics for the growth rate of the number of graded ( 3 + 1 ) -avoiding posets.