We study random composite structures considered up to symmetry that are sampled according to weights on the inner and outer structures. This model may be viewed as an unlabelled version of Gibbs partitions and encompasses multisets of weighted combinatorial objects. We describe a general setting characterized by the formation of a giant component. The collection of small fragments is shown to converge in total variation toward a limit object following a Pólya–Boltzmann distribution.

The Brownian separable permuton is a random probability measure on the unit square, which was introduced by Bassino, Bouvel, Féray, Gerin and Pierrot (2016) as the scaling limit of the diagram of the uniform separable permutation as size grows to infinity. We show that, almost surely, the permuton is the pushforward of the Lebesgue measure on the graph of a random measure-preserving function associated to a Brownian excursion whose strict local minima are decorated with independent and identically distributed signs. As a consequence, its support is almost surely totally disconnected, has Hausdorff dimension one, and enjoys self-similarity properties inherited from those of the Brownian excursion. The density function of the averaged permuton is computed and a connection with the shuffling of the Brownian continuum random tree is explored.

The $W$-operator, $W([n])$, generalises the cut-and-join operator. We prove that $W([n])$ can be written as the sum of $n!$ terms, each term corresponding uniquely to a permutation in $S_{\!n}$. We also prove that there is a correspondence between the terms of $W([n])$ with maximal degree and noncrossing partitions.

For $$\tau \in {S_3}$$, let $$\mu _n^\tau $$ denote the uniformly random probability measure on the set of $$\tau $$-avoiding permutations in $${S_n}$$. Let $${\mathbb {N}^*} = {\mathbb {N}} \cup \{ \infty \} $$ with an appropriate metric and denote by $$S({\mathbb{N}},{\mathbb{N}^*})$$ the compact metric space consisting of functions $$\sigma {\rm{= }}\{ {\sigma _i}\} _{i = 1}^\infty {\rm{}}$$ from $$\mathbb {N}$$ to $${\mathbb {N}^ * }$$ which are injections when restricted to $${\sigma ^{ - 1}}(\mathbb {N})$$; that is, if $${\sigma _i}{\rm{= }}{\sigma _j}$$, $$i \ne j$$, then $${\sigma _i} = \infty $$. Extending permutations $$\sigma \in {S_n}$$ by defining $${\sigma _j} = j$$, for $$j \gt n$$, we have $${S_n} \subset S({\mathbb{N}},{{\mathbb{N}}^*})$$. For each $$\tau \in {S_3}$$, we study the limiting behaviour of the measures $$\{ \mu _n^\tau \} _{n = 1}^\infty $$ on $$S({\mathbb{N}},{\mathbb{N}^*})$$. We obtain partial results for the permutation $$\tau= 321$$ and complete results for the other five permutations $$\tau \in {S_3}$$.

Let $p>3$ be a prime and let $a$ be a rational $p$-adic integer with $a\not \equiv 0\;(\text{mod}\;p)$. We evaluate

$$\begin{eqnarray}\mathop{\sum }_{k=1}^{(p-1)/2}\frac{1}{k}\binom{a}{k}\binom{-1-a}{k}\quad \text{and}\quad \mathop{\sum }_{k=0}^{(p-1)/2}\frac{1}{2k-1}\binom{a}{k}\binom{-1-a}{k}\end{eqnarray}$$

$p^{2}$

We prove a new linear relation for multiple zeta values. This is a natural generalisation of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values.

We generalise a result of Chern [‘A curious identity and its applications to partitions with bounded part differences’, New Zealand J. Math. 47 (2017), 23–26] on distinct partitions with bounded difference between largest and smallest parts. The generalisation is proved both analytically and bijectively.

We study generating functions of ordinary and plane partitions coloured by the action of a finite subgroup of the corresponding special linear group. After reviewing known results for the case of ordinary partitions, we formulate a conjecture concerning a basic factorisation property of the generating function of coloured plane partitions that can be thought of as an orbifold analogue of a conjecture of Maulik et al., now a theorem, in three-dimensional Donaldson–Thomas theory. We study natural quantisations of the generating functions arising from geometry, discuss a quantised version of our conjecture, and prove a positivity result for the quantised coloured plane partition function under a geometric assumption.

Let $n$ be a positive integer and $a$ an integer prime to $n$. Multiplication by $a$ induces a permutation over $\mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$. Lerch’s theorem gives the sign of this permutation. We explore some applications of Lerch’s result to permutation problems involving quadratic residues modulo $p$ and confirm some conjectures posed by Sun [‘Quadratic residues and related permutations and identities’, Preprint, 2018, arXiv:1809.07766]. We also study permutations involving arbitrary $k$th power residues modulo $p$ and primitive roots modulo a power of $p$.

We completely classify Cartan subalgebras of dimension drop algebras with coprime parameters. More generally, we classify Cartan subalgebras of arbitrary stabilised dimension drop algebras that are non-degenerate in the sense that the dimensions of their fibres in the endpoints are maximal. Conjugacy classes by an automorphism are parametrised by certain congruence classes of matrices over the natural numbers with prescribed row and column sums. In particular, each dimension drop algebra admits only finitely many non-degenerate Cartan subalgebras up to conjugacy. As a consequence of this parametrisation, we can provide examples of subhomogeneous $\text{C}^{\ast }$-algebras with exactly $n$ Cartan subalgebras up to conjugacy. Moreover, we show that in many dimension drop algebras two Cartan subalgebras are conjugate if and only if their spectra are homeomorphic.

In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree is unimodal. Although it attracts many researchers' attention, it is still open. Motivated by this conjecture, in this paper, we prove that rooted products of some graphs preserve real rootedness of independence polynomials. As application, we not only give a unified proof for some known results, but also we can apply them to generate infinite kinds of trees whose independence polynomials have only real zeros. Thus their independence polynomials are unimodal.

For positive integers $t_{1},\ldots ,t_{k}$, let $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ (respectively $p(n,t_{1},t_{2},\ldots ,t_{k})$) be the number of partitions of $n$ such that, if $m$ is the smallest part, then each of $m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$ appears as a part and the largest part is at most (respectively equal to) $m+t_{1}+t_{2}+\cdots +t_{k}$. Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of $p(n,t_{1},t_{2},\ldots ,t_{k})$. We establish a $q$-series identity from which the formulae for the generating functions of $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ and $p(n,t_{1},t_{2},\ldots ,t_{k})$ can be obtained.

Let (X, Y) = (Xn, Yn)n≥1 be the output process generated by a hidden chain Z = (Zn)n≥1, where Z is a finite-state, aperiodic, time homogeneous, and irreducible Markov chain. Let LCn be the length of the longest common subsequences of X1,..., Xn and Y1,..., Yn. Under a mixing hypothesis, a rate of convergence result is obtained for E[LCn]/n.

Our main result establishes Andrews’ conjecture for the asymptotic of the generating function for the number of integer partitions of $n$ without $k$ consecutive parts. The methods we develop are applicable in obtaining asymptotics for stochastic processes that avoid patterns; as a result they yield asymptotics for the number of partitions that avoid patterns.

Holroyd, Liggett, and Romik, in connection with certain bootstrap percolation models, introduced the study of partitions without $k$ consecutive parts. Andrews showed that when $k=2$, the generating function for these partitions is a mixed-mock modular form and, thus, has modularity properties which can be utilized in the study of this generating function. For $k>2$, the asymptotic properties of the generating functions have proved more difficult to obtain. Using $q$-series identities and the $k=2$ case as evidence, Andrews stated a conjecture for the asymptotic behavior. Extensive computational evidence for the conjecture in the case $k=3$ was given by Zagier.

This paper improved upon early approaches to this problem by identifying and overcoming two sources of error. Since the writing of this paper, a more precise asymptotic result was established by Bringmann, Kane, Parry, and Rhoades. That approach uses very different methods.

We prove that, given a positive integer $m$, there is a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that

$$\begin{eqnarray}m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots +\frac{1}{n_{k}}\end{eqnarray}$$

$\{1/n_{i}\}_{i=1}^{k}$

We construct a shifted version of the Turán sieve method developed by R. Murty and the second author and apply it to counting problems on tournaments. More precisely, we obtain upper bounds for the number of tournaments which contain a fixed number of restricted $r$-cycles. These are the first concrete results which count the number of cycles over “all tournaments”.

A cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{k}(x)$ is an essential cyclotomic factor of $f(x)\in \mathbb{Z}[x]$ if $\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$ and every prime divisor of $k$ is less than or equal to the number of terms of $f.$ We show that if a monic polynomial with coefficients from $\{-1,0,1\}$ has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].

Given a sequence of polynomials $\{x_k(q)\}_{k \ges 0}$, define the transformation

$$y_n(q) = a^n\sum\limits_{i = 0}^n {\left( \matrix{n \cr i} \right)} b^{n-i}x_i(q)$$

$n\ges 0$

$[x_{i+j}(q)]_{i,j \ges 0}$

$\{y_n(q)\}_{n\ges 0}$

Bevan established that the growth rate of a monotone grid class of permutations is equal to the square of the spectral radius of a related bipartite graph. We give an elementary and self-contained proof of a generalization of this result using only Stirling's formula, the method of Lagrange multipliers, and the singular value decomposition of matrices. Our proof relies on showing that the maximum over the space of n × n matrices with non-negative entries summing to one of a certain function of those entries, parametrized by the entries of another matrix Γ of non-negative real numbers, is equal to the square of the largest singular value of Γ and that the maximizing point can be expressed as a Hadamard product of Γ with the tensor product of singular vectors for its greatest singular value.

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 $n$.