Weighted partition rank and crank moments. III. A list of Andrews–Beck type congruences modulo 5, 7, 11 and 13

Let [Formula: see text] count the total number of parts among partitions of [Formula: see text] with rank congruent to [Formula: see text] modulo [Formula: see text] and let [Formula: see text] count the total appearances of ones among partitions of [Formula: see text] with crank congruent to [Formula: see text] modulo [Formula: see text]. We provide a list of over 70 congruences modulo 5, 7, 11 and 13 involving [Formula: see text] and [Formula: see text], which are known as congruences of Andrews–Beck type. Some recent conjectures of Chan, Mao and Osburn are also included in this list.

Hidden Symmetries of Weighted Lozenge Tilings

We study the weighted partition function for lozenge tilings, with weights given by multivariate rational functions originally defined by Morales, Pak and Panova (2019) in the context of the factorial Schur functions. We prove that this partition function is symmetric for large families of regions. We employ both combinatorial and algebraic proofs.

The Size of the Largest Part of Random Weighted Partitions of Large Integers

We consider partitions of the positive integernwhose parts satisfy the following condition. For a given sequence of non-negative numbers {bk}k≥1, a part of sizekappears in exactlybkpossible types. Assuming that a weighted partition is selected uniformly at random from the set of all such partitions, we study the asymptotic behaviour of the largest partXn. LetD(s)=∑k=1∞bkk−s,s=σ+iy, be the Dirichlet generating series of the weightsbk. Under certain fairly general assumptions, Meinardus (1954) obtained the asymptotic of the total number of such partitions asn→∞. Using the Meinardus scheme of conditions, we prove thatXn, appropriately normalized, converges weakly to a random variable having Gumbel distribution (i.e., its distribution function equalse−e−t, −∞<t<∞). This limit theorem extends some known results on particular types of partitions and on the Bose–Einstein model of ideal gas.

Limit Theorems for the Number of Parts in a Random Weighted Partition

Let $c_{m,n}$ be the number of weighted partitions of the positive integer $n$ with exactly $m$ parts, $1\le m\le n$. For a given sequence $b_k, k\ge 1,$ of part type counts (weights), the bivariate generating function of the numbers $c_{m,n}$ is given by the infinite product $\prod_{k=1}^\infty(1-uz^k)^{-b_k}$. Let $D(s)=\sum_{k=1}^\infty b_k k^{-s}, s=\sigma+iy,$ be the Dirichlet generating series of the weights $b_k$. In this present paper we consider the random variable $\xi_n$ whose distribution is given by $P(\xi_n=m)=c_{m,n}/(\sum_{m=1}^nc_{m,n}), 1\le m\le n$. We find an appropriate normalization for $\xi_n$ and show that its limiting distribution, as $n\to\infty$, depends on properties of the series $D(s)$. In particular, we identify five different limiting distributions depending on different locations of the complex half-plane in which $D(s)$ converges.