Let $P$ be a finite nonempty poset with $n$ elements, let $f:P\to\{1,\ldots,n\}$ be a uniformly random order-preserving bijection, and put $h_P(x)=\mathbb E[f(x)]$. Define $\operatorname{gap}(P)$ as the largest difference between consecutive values in the ordered list consisting of $0$, $n+1$, and all the expected ranks $h_P(x)$. Write $w(P)$ for the largest size of a pairwise incomparable subset. We prove three results. The first proves an old conjectural relation between width and expected-rank gaps that has appeared repeatedly, in increasingly general forms, in work of Brightwell and Trotter (2002), Bir\'o and Trotter (2011), and Aires and Kahn (2025): $\operatorname{gap}(P)\le 2w(P)-1$. Second, for every $L>0$ we construct a width-two poset such that every maximal chain has an expected-rank gap of at least $L$, where the two endpoint spacings are included when computing this gap. Finally, for every $r\in\mathbb N$, we construct a poset $P_r$ for which the relative order induced on every nonempty selected set $X$ has base-two entropy below $3|X|$, while $\operatorname{gap}(P_r)\ge(3/2)^r$. Thus the gap can be arbitrarily large while the induced order on every selected set has relatively small entropy. The key ideas behind all three results were found by ChatGPT 5.6 Sol.
We recall the concept of the dimension of a finite poset $P$, and the longstanding conjecture that for all finite nonempty posets $P$ and $Q$, $\dim(P\times Q)\geq\dim(P)+\dim(Q)-2.$ We then note two other plausible inequalities, either of which would imply that one. In the final section, writing $P\preccurlyeq P'$ if, for all $Q,$ $\dim(P\times Q)\leq\dim(P'\times Q),$ and writing $P\approx P'$ if $P\preccurlyeq P'$ and $P'\preccurlyeq P,$ we note some results and questions concerning these relations.
Let $q=p^k$, where $p$ is prime, and let $M$ be a finite matroid representable over ${\Bbb{F}}_q$. Write $\chi_M(t)$ for its characteristic polynomial and $\mbox{decop}(M)$ for the least number of independent sets needed to cover its ground set. We prove that $\chi_M(q)>0$ whenever $k\ge\mbox{decop}(M)$. Geometrically, the central hyperplanes determined by any representation of $M$ fail to cover the dual of the ambient vector space. The proof rests on the Frobenius-power ideals $(X_1^{p^s},\ldots,X_n^{p^s})$, $s\ge1$. Each is preserved by every linear change of coordinates, while nonmembership records the existence of a monomial whose exponent in every variable is bounded. This permits successive normalizations of several invertible systems of linear forms without losing the exponent bounds already obtained. The coefficient form of the Combinatorial Nullstellensatz then produces a common nowhere-zero point. Finally, we test the scope of the theorem. M.~J.~Moghaddamzadeh's unpublished conjecture predicts a stronger statement over prime fields. Projective geometries show that its direct analogue fails over proper extension fields, even under the same numerical inequality.
We prove the $p$-Frobenius problems proposed as Conjectures 7.1 and 7.5 developed by T. Komatsu and R. Pandey (Bull. Korean Math. Soc. 2025;62:1397--1409.) for two families of consecutive odd integers. For integers $r,L,n\ge0$, the bounded restricted partition function $p_{\le r}^{(\le L)}(\le n)$ counts partitions of $n$ into at most $r$ parts, each at most $L$. Thus the bounded restricted partition functions $p_{\le 3}^{(\le a)}(\le s)$and $p_{\le 3}^{(\le a+1)}(\le s)$ play central roles in the proofs. Their generating functions are Gaussian polynomials, whose symmetry and unimodality provide a common tool for treating both families.
Let $h$ be a positive integer, and let $A$ be a finite set of integers. We derive an exact formula for $|hA|$. Furthermore, let $A=\{0,1,\ldots,s,a,b\}$, $1\leq s<a<b$, and write $b=qa+r$ with $0\leq r<a$. By using generating function, we prove that $|hA|$ equals a definite explicit formula expressed in terms of certain truncated binomial coefficients for all positive integers $h$ if and only if $r=0$ or $qs+r\geq a$. This generalizes a result of Nathanson.
Let $H$ be a numerical monoid, that is, a cofinite submonoid of $\mathbb N$ (the non-negative integers under addition). Denote by $\mathcal P_{\text{fin},0}(H)$ the monoid obtained by endowing the family of all finite subsets of $H$ containing $0$ with the operation of setwise addition induced by $H$ on its power set. Tringali and Yan [JCTA, 2025] have recently established that $\mathcal P_{\text{fin},0}(\mathbb N)$ has a unique non-trivial automorphism, and conjectured that the automorphism group of $\mathcal P_{\text{fin},0}(H)$ is trivial whenever $H \ne \mathbb N$. We prove this conjecture and, as a byproduct, give a new proof of the Tringali--Yan theorem.
Let $X_1,X_2,\ldots$ be independent $\mathrm{Bernoulli}(\theta)$ random variables, and let $\bar X_n = n^{-1}(X_1 + \cdots + X_n)$. We prove that, for every real $p \geq 1$, the sequence $\{\mathsf{E}(\bar X_n^p)\}_{n \geq 1}$ is log-convex. This proves the Bernoulli case of a conjecture of Lamkin and Tkocz [Canad. Math. Bull., 65(2):271-278, 2022]. The proof conditions on the total number of successes among $2n$ trials and reduces the desired inequality to a convex-order comparison between two normalized quadratic functions of hypergeometric random variables. All the log-convexity inequalities are strict for $p>1$ and $0<\theta<1$.
