Let $n,r,k\in \mathbb{N}$. An $r$-colouring of the vertices of a regular $n$-gon is any mapping $\unicode[STIX]{x1D712}:\mathbb{Z}_{n}\rightarrow \{1,2,\ldots ,r\}$. Two colourings are equivalent if one of them can be obtained from another by a rotation of the polygon. An $r$-ary necklace of length $n$ is an equivalence class of $r$-colourings of $\mathbb{Z}_{n}$. We say that a colouring is $k$-alternating if all $k$ consecutive vertices have pairwise distinct colours. We compute the smallest number $r$ for which there exists a $k$-alternating $r$-colouring of $\mathbb{Z}_{n}$ and we count, for any $r$, 2-alternating $r$-colourings of $\mathbb{Z}_{n}$ and 2-alternating $r$-ary necklaces of length $n$.

We give combinatorial descriptions of two stochastic growth models for series-parallel networks introduced by Hosam Mahmoud by encoding the growth process via recursive tree structures. Using decompositions of the tree structures and applying analytic combinatorics methods allows a study of quantities in the corresponding series-parallel networks. For both models we obtain limiting distribution results for the degree of the poles and the length of a random source-to-sink path, and furthermore we get asymptotic results for the expected number of source-to-sink paths. Moreover, we introduce generalizations of these stochastic models by encoding the growth process of the networks via further important increasing tree structures.

Athanasiadis [‘A survey of subdivisions and local $h$-vectors’, in The Mathematical Legacy of Richard P. Stanley (American Mathematical Society, Providence, RI, 2017), 39–51] asked whether the local $h$-polynomials of type $A$ cluster subdivisions have only real zeros. We confirm this conjecture and prove that the local $h$-polynomials for all the Cartan–Killing types have only real roots. Our proofs use multiplier sequences and Chebyshev polynomials of the second kind.

A random binary search tree grown from the uniformly random permutation of [n] is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction ck of vertices of a fixed rank k ≥ 0 is shown to decay exponentially with k. We prove that the ranks of the uniformly random, fixed size sample of vertices are asymptotically independent, each having the distribution {ck}. Notoriously hard to compute, the exact fractions ck have been determined for k ≤ 3 only. We present a shortcut enabling us to compute c4 and c5 as well; both are ratios of enormous integers, the denominator of c5 being 274 digits long. Prompted by the data, we prove that, in sharp contrast, the largest prime divisor of the denominator of ck is at most 2k+1 + 1. We conjecture that, in fact, the prime divisors of every denominator for k > 1 form a single interval, from 2 to the largest prime not exceeding 2k+1 + 1.

We prove the unimodality of some coloured $q$-Eulerian polynomials, which involve the flag excedances, the major index and the fixed points on coloured permutation groups, via two recurrence formulas. In particular, we confirm a recent conjecture of Mongelli about the unimodality of the flag excedances over type B derangements. Furthermore, we find the coloured version of Gessel’s hook factorisation, which enables us to interpret these two recurrences combinatorially. We also provide a combinatorial proof of a symmetric and unimodal expansion for the coloured derangement polynomial, which was first established by Shin and Zeng using continued fractions.

Let $\mathcal{F}$ = {F1, F2,. . ., Fn} be a family of n sets on a ground set S, such as a family of balls in ℝd. For every finite measure μ on S, such that the sets of $\mathcal{F}$ are measurable, the classical inclusion–exclusion formula asserts that

$\[\mu(F_1\cup F_2\cup\cdots\cup F_n)=\sum_{I:\emptyset\ne I\subseteq[n]} (-1)^{|I|+1}\mu\biggl(\bigcap_{i\in I} F_i\biggr),\]$

$\mathcal{F}$

$\mathcal{F}$

$\mathcal{F}$

This paper concerns a relatively new combinatorial structure called staircase tableaux. They were introduced in the context of the asymmetric exclusion process and Askey–Wilson polynomials; however, their purely combinatorial properties have gained considerable interest in the past few years.

In this paper we further study combinatorial properties of staircase tableaux. We consider a general model of random staircase tableaux in which symbols (Greek letters) that appear in staircase tableaux may have arbitrary positive weights. (We consider only the case with the parameters u = q = 1.) Under this general model we derive a number of results. Some of our results concern the limiting laws for the number of appearances of symbols in a random staircase tableaux. They generalize and subsume earlier results that were obtained for specific values of the weights.

One advantage of our generality is that we may let the weights approach extreme values of zero or infinity, which covers further special cases appearing earlier in the literature. Furthermore, our generality allows us to analyse the structure of random staircase tableaux, and we obtain several results in this direction.

One of the tools we use is the generating functions of the parameters of interest. This leads us to a two-parameter family of polynomials, generalizing the classical Eulerian polynomials.

We also briefly discuss the relation of staircase tableaux to the asymmetric exclusion process, to other recently introduced types of tableaux, and to an urn model studied by a number of researchers, including Philippe Flajolet.

We define the notion of smooth supercritical compositional structures. Two well-known examples are compositions and graphs of given genus. The ‘parts’ of a graph are the subgraphs that are maximal trees. We show that large part sizes have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. In many cases this leads to asymptotic formulas for the probability of being gap-free and for the expected values of the largest part sizes, number of distinct parts and number of parts of multiplicity k.

In this paper, we show that the $\gamma $-vectors of Coxeter complexes (of types A and B) and associahedrons (of types A and B) can be obtained by using derivative polynomials of the tangent and secant functions. We provide a unified grammatical approach to generate these $\gamma $-vectors and the coefficient arrays of Narayana polynomials, Legendre polynomials and Chebyshev polynomials of both kinds.