Counting minimal generator matrices

Given a particular convolutional code C, we wish to find all minimal generator matrices G(D) which represent that code. A standard form S(D) for a minimal matrix is defined, and then all standard forms for the code C are counted (this is equivalent to counting special pre-multiplication matrices P(D)). It is shown that all the minimal generator matrices G(D) are contained within the 'ordered row permutations' of these standard forms, and that all these permutations are distinct. Finally, the result is used to place a simple upper bound on the possible number of convolutional codes

Enumeration of permutations by the parity of descent position

Noticing that some recent variations of descent polynomials are special cases of Carlitz and Scoville's four-variable polynomials, which enumerate permutations by the parity of descent and ascent position, we prove a q-analogue of Carlitz-Scoville's generating function by counting the inversion number and a type B analogue by enumerating the signed permutations with respect to the parity of desecnt and ascent position. As a by-product of our formulas, we obtain a q-analogue of Chebikin's formula for alternating descent polynomials, an alternative proof of Sun's gamma-positivity of her bivariate Eulerian polynomials and a type B analogue, the latter refines Petersen's gamma-positivity of the type B Eulerian polynomials.Comment: 26 page

Congruence successions in compositions

A \emph{composition} is a sequence of positive integers, called \emph{parts}, having a fixed sum. By an \emph{$m$-congruence succession}, we will mean a pair of adjacent parts $x$ and $y$ within a composition such that $x\equiv y(\text{mod} m)$. Here, we consider the problem of counting the compositions of size $n$ according to the number of $m$-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case $m=2$, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size $n$ having no $m$-congruence successions

Enumerative combinatorics, continued fractions and total positivity

Determining whether a given number is positive is a fundamental question in mathematics. This can sometimes be answered by showing that the number counts some collection of objects, and hence, must be positive. The work done in this dissertation is in the field of enumerative combinatorics, the branch of mathematics that deals with exact counting. We will consider several problems at the interface between enumerative combinatorics, continued fractions and total positivity. In our first contribution, we exhibit a lower-triangular matrix of polynomials in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. This generalises Brenti’s conjecture from 1996. We prove the coefficientwise total positivity of a three-variable case which includes the reversed Stirling subset triangle. Our next contribution is the study of two sequences whose Stieltjes-type continued fraction coefficients grow quadratically; we study the Genocchi and median Genocchi numbers. We find Stieltjes-type and Thron-type continued fractions for some multivariate polynomials that enumerate D-permutations, a class of permutations of 2n, with respect to a very large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings. After this, we interpret the Foata–Zeilberger bijection in terms of Laguerre digraphs, which enables us to count cycles in permutations. Using this interpretation, we obtain Jacobi-type continued fractions for multivariate polynomials enumerating permutations, and also Thron-type and Stieltjes-type continued fractions for multivariate polynomials enumerating D-permutations, in both cases including the counting of cycles. This enables us to prove some conjectured continued fractions due to Sokal–Zeng from 2022, and Randrianarivony–Zeng from 1996. Finally, we introduce the higher-order Stirling cycle and subset numbers; these generalise the Stirling cycle and subset numbers, respectively. We introduce some conjectures which involve different total-positivity questions for these triangular arrays and then answer some of them

On the refined counting of graphs on surfaces

Ribbon graphs embedded on a Riemann surface provide a useful way to describe the double line Feynman diagrams of large N computations and a variety of other QFT correlator and scattering amplitude calculations, e.g in MHV rules for scattering amplitudes, as well as in ordinary QED. Their counting is a special case of the counting of bi-partite embedded graphs. We review and extend relevant mathematical literature and present results on the counting of some infinite classes of bi-partite graphs. Permutation groups and representations as well as double cosets and quotients of graphs are useful mathematical tools. The counting results are refined according to data of physical relevance, such as the structure of the vertices, faces and genus of the embedded graph. These counting problems can be expressed in terms of observables in three-dimensional topological field theory with S_d gauge group which gives them a topological membrane interpretation.Comment: 57 pages, 12 figures; v2: Typos corrected; references adde

Genus Permutations and Genus Partitions

For a given permutation or set partition there is a natural way to associate a genus. Counting all permutations or partitions of a fixed genus according to cycle lengths or block sizes, respectively, is the main content of this article. After a variable transformation, the generating series are rational functions with poles located at the ramification points in the new variable. The generating series for any genus is given explicitly for permutations and up to genus 2 for set partitions. Extending the topological structure not just by the genus but also by adding more boundaries, we derive the generating series of non-crossing partitions on the cylinder from known results of non-crossing permutations on the cylinder. Most, but not all, outcomes of this article are special cases of already known results, however they are not represented in this way in the literature, which however seems to be the canonical way. To make the article as accessible as possible, we avoid going into details into the explicit connections to Topological Recursion and Free Probability Theory, where the original motivation came from.Comment: 27 pages, 4 figures, comments are appreciate