The payoff in the Chow–Robbins coin-tossing game is the proportion of heads when you stop. Stopping to maximize expectation was addressed by Chow and Robbins (1965), who proved there exist integers ${k_n}$ such that it is optimal to stop at n tosses when heads minus tails is ${k_n}$. Finding ${k_n}$ was unsolved except for finitely many cases by computer. We prove an $o(n^{-1/4})$ estimate of the stopping boundary of Dvoretsky (1967), which then proves ${k_n} = \left\lceil {\alpha \sqrt n \,\, - 1/2\,\, + \,\,\frac{{\left( { - 2\zeta (\! -1/2)} \right)\sqrt \alpha }}{{\sqrt \pi }}{n^{ - 1/4}}} \right\rceil $ except for n in a set of density asymptotic to 0, at a power law rate. Here, $\alpha$ is the Shepp–Walker constant from the Brownian motion analog, and $\zeta$ is Riemann’s zeta function. An $n^{ - 1/4}$ dependence was conjectured by Christensen and Fischer (2022). Our proof uses moments involving Catalan and Shapiro Catalan triangle numbers which appear in a tree resulting from backward induction, and a generalized backward induction principle. It was motivated by an idea of Häggström and Wästlund (2013) to use backward induction of upper and lower Value bounds from a horizon, which they used numerically to settle a few cases. Christensen and Fischer, with much better bounds, settled many more cases. We use Skorohod’s embedding to get simple upper and lower bounds from the Brownian analog; our upper bound is the one found by Christensen and Fischer in another way. We use them first for yet many more examples and a conjecture, then algebraically in the tree, with feedback to get much sharper Value bounds near the border, and analytic results. Also, we give a formula that gives the exact optimal stop rule for all n up to about a third of a billion; it uses the analytic result plus terms arrived at empirically.

Amdeberhan et al. [‘Arithmetic properties for generalized cubic partitions and overpartitions modulo a prime’, Aequationes Math. (2024), doi:10.1007/s00010-024-01116-7] defined the generalised cubic partition function $a_c(n)$ as the number of partitions of n whose even parts may appear in $c\geq 1$ different colours and proved that $a_3(7n+4)\equiv 0\pmod {7}$ and $a_5(11n+10)\equiv 0\pmod {11}$ for all $n\geq 0$ via modular forms. Recently, the author [‘A note on congruences for generalized cubic partitions modulo primes’, Integers 25 (2025), Article no. A20] gave elementary proofs of these congruences. We prove in this note two infinite families of congruences modulo $5$ for $a_c(n)$ given by

$$ \begin{align*} a_{25c+4}(25n+18)&\equiv 0\pmod{5},\\ a_{25c+9}(25n+8)&\equiv 0\pmod{5} \end{align*} $$

for all $c\geq 0$ and $n\geq 0$ using elementary q-series manipulations.

In this work, we investigate the arithmetic properties of $b_{5^k}(n)$, which counts the partitions of n where no part is divisible by $5^k$. By constructing generating functions for $b_{5^k}(n)$ across specific arithmetic progressions, we establish a set of Ramanujan-type congruences.

Andrews and El Bachraoui [‘On two-colour partitions with odd smallest part’, Preprint, arXiv:2410.14190] explored many integer partitions in two colours, some of which are generated by the mock theta functions of third order of Ramanujan and Watson. They also posed questions regarding combinatorial proofs for these results. In this paper, we establish bijections to provide a combinatorial proof of one of these results and a companion result. We give analytic proofs of further companion results.

We establish two new truncated identities for theta series. Their partition-theoretic interpretations are also given.

Inspired by work of Andrews and Newman [‘Partitions and the minimal excludant’, Ann. Comb. 23 (2019), 249–254] on the minimal excludant or ‘mex’ of partitions, we define four new classes of minimal excludants for overpartitions and establish relations to certain functions due to Ramanujan.

A pebble tree is an ordered tree where each node receives some colored pebbles, in such a way that each unary node receives at least one pebble, and each subtree has either one more or as many leaves as pebbles of each color. We show that the contraction poset on pebble trees is isomorphic to the face poset of a convex polytope called pebble tree polytope. Beside providing intriguing generalizations of the classical permutahedra and associahedra, our motivation is that the faces of the pebble tree polytopes provide realizations as convex polytopes of all assocoipahedra constructed by K. Poirier and T. Tradler only as polytopal complexes.

In his “lost notebook,” Ramanujan used iterated derivatives of two theta functions to define sequences of q-series $\{U_{2t}(q)\}$ and $\{V_{2t}(q)\}$ that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms of “partition Eisenstein series,” extensions of the classical Eisenstein series $E_{2k}(q),$ defined by

$$ \begin{align*}\lambda=(1^{m_1}, 2^{m_2},\dots, n^{m_n}) \vdash n \ \ \ \longmapsto \ \ \ E_{\lambda}(q):= E_2(q)^{m_1} E_4(q)^{m_2}\cdots E_{2n}(q)^{m_n}. \end{align*} $$

For functions $\phi : \mathcal {P}\mapsto {\mathbb C}$ on partitions, the weight $2n$ partition Eisenstein trace is

$$ \begin{align*}\operatorname{\mathrm{Tr}}_n(\phi;q):=\sum_{\lambda \vdash n} \phi(\lambda)E_{\lambda}(q). \end{align*} $$

For all t, we prove that $U_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _U;q)$ and $V_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _V;q),$ where $\phi _U$ and $\phi _V$ are natural partition weights, giving the first explicit quasimodular formulas for these series.

For $M,N,m\in \mathbb {N}$ with $M\geq 2, N\geq 1$ and $m \geq 2$, we define two families of sequences, $\mathcal {B}_{M,N}$ and $\mathcal {B}_{M,N,m}$. The nth term of $\mathcal {B}_{M,N}$ is obtained by expressing n in base M and recombining those digits in base N. The nth term of $\mathcal {B}_{M,N,m}$ is defined by $\mathcal {B}_{M,N,m}(n):=\mathcal {B}_{M,N}(n) \; (\textrm {mod}\;m)$. The special case $\mathcal {B}_{M,1,m}$, where $N=1$, yields the digit sum sequence $\mathbf {t}_{M,m}$ in base M mod m. We prove that $\mathcal {B}_{M,N}$ is the fixed point of a morphism $\mu _{M,N}$ at letter $0$, similar to a property of $\mathbf {t}_{M,m}$. Additionally, we show that $\mathcal {B}_{M,N,m}$ contains arbitrarily long palindromes if and only if $m=2$, mirroring the behaviour of the digit sum sequence. When $m\geq M$ and N, m are coprime, we establish that $\mathcal {B}_{M,N,m}$ contains no overlaps.

Let $\overline {M}(a,c,n)$ denote the number of overpartitions of n with first residual crank congruent to a modulo c with $c\geq 3$ being odd and $0\leq a<c$. The central objective of this paper is twofold: firstly, to establish an asymptotic formula for the crank of overpartitions; and secondly, to establish several inequalities concerning $\overline {M}(a,c,n)$ that encompasses crank differences, positivity, and strict log-subadditivity.

In his proof of the irrationality of $\zeta (3)$ and $\zeta (2)$, Apéry defined two integer sequences through $3$-term recurrences, which are known as the famous Apéry numbers. Zagier, Almkvist–Zudilin, and Cooper successively introduced the other $13$ sporadic sequences through variants of Apéry’s $3$-term recurrences. All of the $15$ sporadic sequences are called Apéry-like sequences. Motivated by Gessel’s congruences mod $24$ for the Apéry numbers, we investigate congruences of the form $u_n\equiv \alpha ^n \ \pmod {N_{\alpha }}~(\alpha \in \mathbb {Z},N_{\alpha }\in \mathbb {N}^{+})$ for all of the $15$ Apéry-like sequences $\{u_n\}_{n\ge 0}$. Let $N_{\alpha }$ be the largest positive integer such that $u_n\equiv \alpha ^n\ \pmod {N_{\alpha }}$ for all non-negative integers n. We determine the values of $\max \{N_{\alpha }|\alpha \in \mathbb {Z}\}$ for all of the $15$ Apéry-like sequences $\{u_n\}_{n\ge 0}$. The binomial transforms of Apéry-like sequences provide us a unified approach to this type of congruences for Apéry-like sequences.

A classical result of Erdős, Lovász and Spencer from the late 1970s asserts that the dimension of the feasible region of densities of graphs with at most k vertices in large graphs is equal to the number of non-trivial connected graphs with at most k vertices. Indecomposable permutations play the role of connected graphs in the realm of permutations, and Glebov et al. showed that pattern densities of indecomposable permutations are independent, i.e., the dimension of the feasible region of densities of permutation patterns of size at most k is at least the number of non-trivial indecomposable permutations of size at most k. However, this lower bound is not tight already for $k=3$. We prove that the dimension of the feasible region of densities of permutation patterns of size at most k is equal to the number of non-trivial Lyndon permutations of size at most k. The proof exploits an interplay between algebra and combinatorics inherent to the study of Lyndon words.

An identity that is reminiscent of the Littlewood identity plays a fundamental role in recent proofs of the facts that alternating sign triangles are equinumerous with totally symmetric self-complementary plane partitions and that alternating sign trapezoids are equinumerous with holey cyclically symmetric lozenge tilings of a hexagon. We establish a bounded version of a generalization of this identity. Further, we provide combinatorial interpretations of both sides of the identity. The ultimate goal would be to construct a combinatorial proof of this identity (possibly via an appropriate variant of the Robinson-Schensted-Knuth correspondence) and its unbounded version, as this would improve the understanding of the mysterious relation between alternating sign trapezoids and plane partition objects.

Recently, Alanazi et al. [‘Refining overpartitions by properties of nonoverlined parts’, Contrib. Discrete Math. 17(2) (2022), 96–111] considered overpartitions wherein the nonoverlined parts must be $\ell $-regular, that is, the nonoverlined parts cannot be divisible by the integer $\ell $. In the process, they proved a general parity result for the corresponding enumerating functions. They also proved some specific congruences for the case $\ell =3$. In this paper we use elementary generating function manipulations to significantly extend this set of known congruences for these functions.

We give an explicit formula for the Frobenius number of triples associated with the Diophantine equation $x^2+y^2=z^3$, that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of $x^2+y^2=z^3$. For the equation $x^2+y^2=z^2$, the Frobenius number has already been given. Our approach can be extended to the general equation $x^2+y^2=z^r$ for $r>3$.

Describing the equality conditions of the Alexandrov–Fenchel inequality [Ale37] has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem and is a complexity counterpart of the recent result by Shenfeld and van Handel [SvH23], which gave a geometric characterization of the equality conditions. The proof involves Stanley’s [Sta81] order polytopes and employs poset theoretic technology.

Given a permutation statistic $\operatorname {\mathrm {st}}$, define its inverse statistic $\operatorname {\mathrm {ist}}$ by . We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of $\operatorname {\mathrm {st}}_{1}$ and $\operatorname {\mathrm {ist}}_{2}$ whenever $\operatorname {\mathrm {st}}_{1}$ and $\operatorname {\mathrm {st}}_{2}$ are descent statistics: permutation statistics that depend only on the descent composition. We apply this method to a number of descent statistics, including the descent number, the peak number, the left peak number, the number of up-down runs and the major index. Perhaps surprisingly, in many cases the polynomial giving the joint distribution of $\operatorname {\mathrm {st}}_{1}$ and $\operatorname {\mathrm {ist}}_{2}$ can be expressed as a simple sum involving products of the polynomials giving the (individual) distributions of $\operatorname {\mathrm {st}}_{1}$ and $\operatorname {\mathrm {st}}_{2}$. Our work leads to a rederivation of Stanley’s generating function for doubly alternating permutations, as well as several conjectures concerning real-rootedness and $\gamma $-positivity.

We study the counts of smooth permutations and smooth polynomials over finite fields. For both counts we prove an estimate with an error term that matches the error term found in the integer setting by de Bruijn more than 70 years ago. The main term is the usual Dickman $\rho$ function, but with its argument shifted.

We determine the order of magnitude of $\log(p_{n,m}/\rho(n/m))$ where $p_{n,m}$ is the probability that a permutation on n elements, chosen uniformly at random, is m-smooth.

We uncover a phase transition in the polynomial setting: the probability that a polynomial of degree n in $\mathbb{F}_q$ is m-smooth changes its behaviour at $m\approx (3/2)\log_q n$.

Given an $n\times n$ symmetric matrix $W\in [0,1]^{[n]\times [n]}$, let ${\mathcal G}(n,W)$ be the random graph obtained by independently including each edge $jk\in \binom{[n]}{2}$ with probability $W_{jk}=W_{kj}$. Given a degree sequence $\textbf{d}=(d_1,\ldots, d_n)$, let ${\mathcal G}(n,\textbf{d})$ denote a uniformly random graph with degree sequence $\textbf{d}$. We couple ${\mathcal G}(n,W)$ and ${\mathcal G}(n,\textbf{d})$ together so that asymptotically almost surely ${\mathcal G}(n,W)$ is a subgraph of ${\mathcal G}(n,\textbf{d})$, where $W$ is some function of $\textbf{d}$. Let $\Delta (\textbf{d})$ denote the maximum degree in $\textbf{d}$. Our coupling result is optimal when $\Delta (\textbf{d})^2\ll \|\textbf{d}\|_1$, that is, $W_{ij}$ is asymptotic to ${\mathbb P}(ij\in{\mathcal G}(n,\textbf{d}))$ for every $i,j\in [n]$. We also have coupling results for $\textbf{d}$ that are not constrained by the condition $\Delta (\textbf{d})^2\ll \|\textbf{d}\|_1$. For such $\textbf{d}$ our coupling result is still close to optimal, in the sense that $W_{ij}$ is asymptotic to ${\mathbb P}(ij\in{\mathcal G}(n,\textbf{d}))$ for most pairs $ij\in \binom{[n]}{2}$.

Let n be a positive integer and $\underline {n}=\{1,2,\ldots ,n\}$. A conjecture arising from certain polynomial near-ring codes states that if $k\geq 1$ and $a_{1},a_{2},\ldots ,a_{k}$ are distinct positive integers, then the symmetric difference $a_{1}\underline {n}\mathbin {\Delta }a_{2}\underline {n}\mathbin {\Delta }\cdots \mathbin {\Delta }a_{k}\underline {n}$ contains at least n elements. Here, $a_{i}\underline {n}=\{a_{i},2a_{i},\ldots ,na_{i}\}$ for each i. We prove this conjecture for arbitrary n and for $k=1,2,3$.