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Register a new userGap between the largest and smallest parts of partitions and Berkovich and Uncus conjectures
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We prove three main conjectures of Berkovich and Uncu (Ann. Comb. 23 (2019) 263--284) on the inequalities between the numbers of partitions of $n$ with bounded gap between largest and smallest parts for sufficiently large $n$. Actually our theorems are stronger than their original conjectures. The analytic version of our results shows that the coefficients of some partition $q$-series are eventually positive.
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For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures -- one regarding the maximum possible value of $cp(G) + cp(overline{G})$ (due to de Caen, ErdH{o}s, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $max_G cp(G) + cp(overline{G})$, and we prove an asymptotic version of the second, showing that $bp_k(K_n) = (1+o(1))n$.
Recently, Deutsch and Elizalde studied the largest and the smallest fixed points of permutations. Motivated by their work, we consider the analogous problems in set partitions. Let $A_{n,k}$ denote the number of partitions of ${1,2,dots, n+1}$ with the largest singleton ${k+1}$ for $0leq kleq n$. In this paper, several explicit formulas for $A_{n,k}$, involving a Dobinski-type analog, are obtained by algebraic and combinatorial methods, many combinatorial identities involving $A_{n,k}$ and Bell numbers are presented by operator methods, and congruence properties of $A_{n,k}$ are also investigated. It will been showed that the sequences $(A_{n+k,k})_{ngeq 0}$ and $(A_{n+k,k})_{kgeq 0}$ (mod $p$) are periodic for any prime $p$, and contain a string of $p-1$ consecutive zeroes. Moreover their minimum periods are conjectured to be $N_p=frac{p^p-1}{p-1}$ for any prime $p$.
Recently, Deutsch and Elizalde studied the largest and the smallest fixed points of permutations. Motivated by their work, we consider the analogous problems in weighted set partitions. Let $A_{n,k}(mathbf{t})$ denote the total weight of partitions on $[n+1]$ with the largest singleton ${k+1}$. In this paper, explicit formulas for $A_{n,k}(mathbf{t})$ and many combinatorial identities involving $A_{n,k}(mathbf{t})$ are obtained by umbral operators and combinatorial methods. As applications, we investigate three special cases such as permutations, involutions and labeled forests. Particularly in the permutation case, we derive a surprising identity analogous to the Riordan identity related to tree enumerations, namely, begin{eqnarray*} sum_{k=0}^{n}binom{n}{k}D_{k+1}(n+1)^{n-k} &=& n^{n+1}, end{eqnarray*} where $D_{k}$ is the $k$-th derangement number or the number of permutations of ${1,2,dots, k}$ with no fixed points.
In this note we show that for each Latin square $L$ of order $ngeq 2$, there exists a Latin square $L eq L$ of order $n$ such that $L$ and $L$ differ in at most $8sqrt{n}$ cells. Equivalently, each Latin square of order $n$ contains a Latin trade of size at most $8sqrt{n}$. We also show that the size of the smallest defining set in a Latin square is $Omega(n^{3/2})$. %That is, there are constants $c$ and $n_0$ such that for any $n>n_0$ the size of the smallest defining %set of order $n$ is at least $cn^{3/2}$.
We define the notion of asymptotically free for locally restricted compositions, which means roughly that large parts can often be replaced by any larger parts. Two well-known examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based on this we obtain asymptotic formulas for the probability of being gap free and for the expected values of the largest part, number of distinct parts and number of parts of multiplicity k, all accurate to o(1).
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