A set S of permutations is forcing if for any sequence $\{\Pi_i\}_{i \in \mathbb{N}}$ of permutations where the density $d(\pi,\Pi_i)$ converges to $\frac{1}{|\pi|!}$ for every permutation $\pi \in S$, it holds that $\{\Pi_i\}_{i \in \mathbb{N}}$ is quasirandom. Graham asked whether there exists an integer k such that the set of all permutations of order k is forcing; this has been shown to be true for any $k\ge 4$. In particular, the set of all 24 permutations of order 4 is forcing. We provide the first non-trivial lower bound on the size of a forcing set of permutations: every forcing set of permutations (with arbitrary orders) contains at least four permutations.

For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality $r-k$. However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W-Narayana numbers.

Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution.

Let $p=3n+1$ be a prime with $n\in \mathbb {N}=\{0,1,2,\ldots \}$ and let $g\in \mathbb {Z}$ be a primitive root modulo p. Let $0<a_1<\cdots <a_n<p$ be all the cubic residues modulo p in the interval $(0,p)$. Then clearly the sequence $a_1 \bmod p,\, a_2 \bmod p,\ldots , a_n \bmod p$ is a permutation of the sequence $g^3 \bmod p,\,g^6 \bmod p,\ldots , g^{3n} \bmod p$. We determine the sign of this permutation.

Gireesh and Mahadeva Naika [‘On 3-regular partitions in 3-colors’, Indian J. Pure Appl. Math. 50 (2019), 137–148] proved an infinite family of congruences modulo powers of 3 for the function $p_{\{3,3\}}(n)$, the number of 3-regular partitions in three colours. In this paper, using elementary generating function manipulations and classical techniques, we significantly extend the list of proven arithmetic properties satisfied by $p_{\{3,3\}}(n).$

We prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$. Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log \log n$ with the prime between $\log n$ and $(\log n)^{\log \log n}$. The proportion of even permutations with this property is at least $1-7/\log \log n$.

Necklaces are the equivalence classes of words under the action of the cyclic group. Let a transition in a word be any change between two adjacent letters modulo the word’s length. We present a closed-form solution for the enumeration of necklaces in n beads, k colours and t transitions. We show that our result provides a more general solution to the problem of counting alternating (proper) colourings of the vertices of a regular n-gon.

We show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for integer moments (after neglecting the first $p^2$ terms of the sequence).

The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver’s inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilised to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g. the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity-preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikvári, which settled Friedland’s lower matching conjecture.

For an indifference graph G, we define a symmetric function of increasing spanning forests of G. We prove that this symmetric function satisfies certain linear relations, which are also satisfied by the chromatic quasisymmetric function and unicellular $\textrm {LLT}$ polynomials. As a consequence, we give a combinatorial interpretation of the coefficients of the $\textrm {LLT}$ polynomial in the elementary basis (up to a factor of a power of $(q-1)$), strengthening the description given in [4].

Fix positive integers k and n with $k \leq n$. Numbers $x_0, x_1, x_2, \ldots , x_{n - 1}$, each equal to $\pm {1}$, are cyclically arranged (so that $x_0$ follows $x_{n - 1}$) in that order. The problem is to find the product $P = x_0x_1 \cdots x_{n - 1}$ of all n numbers by asking the smallest number of questions of the type $Q_i$: what is $x_ix_{i + 1}x_{i + 2} \cdots x_{i+ k -1}$? (where all the subscripts are read modulo n). This paper studies the problem and some of its generalisations.

We find a new refinement of Fine’s partition theorem on partitions into distinct parts with the minimum part odd. As a consequence, we obtain two companion partition identities. Both analytic and combinatorial proofs are provided.

We give a new method of proof for a result of D. Pierre-Loti-Viaud and P. Boulongne which can be seen as a generalization of a characterization of Poisson law due to Rényi and Srivastava. We also provide explicit formulas, in terms of Bell polynomials, for the moments of the compound distributions occurring in the extended collective model in non-life insurance.

Let S be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t, define the asymptotic density

$${c_t} = \mathop {\lim }\limits_{N \to \infty } {1 \over N}|\{ 0 \le n < N:s(n + t) \ge s(n)\} |.$$

A set of integers is primitive if it does not contain an element dividing another. Let f(n) denote the number of maximum-size primitive subsets of {1,…,2n}. We prove that the limit α = limn→∞f(n)1/n exists. Furthermore, we present an algorithm approximating α with (1 + ε) multiplicative error in N(ε) steps, showing in particular that α ≈ 1.318. Our algorithm can be adapted to estimate the number of all primitive sets in {1,…,n} as well.

We address another related problem of Cameron and Erdős. They showed that the number of sets containing pairwise coprime integers in {1,…n} is between ${2^{\pi (n)}} \cdot {e^{(1/2 + o(1))\sqrt n }}$ and ${2^{\pi (n)}} \cdot {e^{(2 + o(1))\sqrt n }}$. We show that neither of these bounds is tight: there are in fact ${2^{\pi (n)}} \cdot {e^{(1 + o(1))\sqrt n }}$ such sets.

The aim of this article is to provoke discussion concerning arithmetic properties of the function $p_{d}(n)$ counting partitions of a positive integer n into dth powers, where $d\geq 2$. Apart from results concerning the asymptotic behaviour of $p_{d}(n)$, little is known. In the first part of the paper, we prove certain congruences involving functions counting various types of partitions into dth powers. The second part of the paper is experimental and contains questions and conjectures concerning the arithmetic behaviour of the sequence $(p_{d}(n))_{n\in \mathbb {N}}$, based on computations of $p_{d}(n)$ for $n\leq 10^5$ for $d=2$ and $n\leq 10^{6}$ for $d=3, 4, 5$.

We show that there are biases in the number of appearances of the parts in two residue classes in the set of ordinary partitions. More precisely, let $p_{j,k,m} (n)$ be the number of partitions of n such that there are more parts congruent to j modulo m than parts congruent to k modulo m for $m \geq 2$. We prove that $p_{1,0,m} (n)$ is in general larger than $p_{0,1,m} (n)$. We also obtain asymptotic formulas for $p_{1,0,m}(n)$ and $p_{0,1,m}(n)$ for $m \geq 2$.

Andrews introduced the partition function $\overline {C}_{k, i}(n)$, called the singular overpartition function, which counts the number of overpartitions of n in which no part is divisible by k and only parts $\equiv \pm i\pmod {k}$ may be overlined. We prove that $\overline {C}_{6, 2}(n)$ is almost always divisible by $2^k$ for any positive integer k. We also prove that $\overline {C}_{6, 2}(n)$ and $\overline {C}_{12, 4}(n)$ are almost always divisible by $3^k$. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of $2$ satisfied by $\overline {C}_{6, 2}(n)$.

We investigate the sum of the parts in all the partitions of n into distinct parts and give two infinite families of linear inequalities involving this sum. The results can be seen as new connections between partitions and divisors.

One of the oldest outstanding problems in dynamical algebraic combinatorics is the following conjecture of P. Cameron and D. Fon-Der-Flaass (1995): consider a plane partition P in an $a \times b \times c$ box ${\sf B}$. Let $\Psi (P)$ denote the smallest plane partition containing the minimal elements of ${\sf B} - P$. Then if $p= a+b+c-1$ is prime, Cameron and Fon-Der-Flaass conjectured that the cardinality of the $\Psi $-orbit of P is always a multiple of p.

This conjecture was established for $p \gg 0$ by Cameron and Fon-Der-Flaass (1995) and for slightly smaller values of p in work of K. Dilks, J. Striker and the second author (2017). Our main theorem specializes to prove this conjecture in full generality.

We provide a generalised Laplace expansion for the permanent function and, as a consequence, we re-prove a multinomial Vandermonde convolution. Some combinatorial identities are derived by applying special matrices to the expansion.