Palmer provides a method of enumerating $n$-plexes, however it has some typographical errors in the formula for the cycle index $Z(S_p^{(r)})$ and the values of $s_p^n$, the number of $n$-plexes on $p$ points. This article is intended to provide the correct formulas.
We shall investigate and arrive at a certain functional property of the double series \[ \sum\limits_{n,r\geq 1}\frac{1}{\sqrt{x^2n^2+r^2+w^2}\left( e^{2 πy\sqrt{x^2n^2+r^2+w^2}}-1\right)}. \]
We complete the classification of B-facets of a 4-dimensional Newton polyhedron, filling a gap in the classification of arXiv:1309.0630, found by the authors of arXiv:2209.03553.
In 2018, Alexander A. Razborov proved that the edge density of Fon-der-Flaass $(3,4)$-graph is $\geq\frac{7}{16}(1-o(1))$, using flag algebras. In this paper, we give an elementary proof of this result.
An arc in $\Z^2_n$ is defined to be a set of points no three of which are collinear. We describe some properties of arcs and determine the maximum size of arcs for some small $n$.
We offer streamlined proofs of fundamental theorems regarding the index theory for partial self-maps of an infinite set that are bijective between cofinite subsets.
The reciprocal Pascal matrix has entries $\binom{i+j}{j}^{-1}$. Explicit formullae for its LU-decomposition, the LU-decomposition of its inverse, and some related matrices are obtained. For all results, $q$-analogues are also presented.
The classification of Grassmannian cluster algebras resembles that of regular polygonal tilings. We conjecture that this resemblance may indicate a deeper connection between these seemingly unrelated structures.
We give a simple proof of the "tree-width duality theorem" of Seymour and Thomas that the tree-width of a finite graph is exactly one less than the largest order of its brambles.
We translate Uchimura's identity for the divisor function and whose generalizations into combinatorics of partitions, and give a combinatorial proof of them. As a by-product of their proofs, we obtain some combinatorial results.
We prove that for any graph G at least one of G or $\bar{G}$ satisfies $χ\leq {1/4}ω+ {3/4}Δ+ 1$. In particular, self-complementary graphs satisfy this bound.