Sum of elements preceding records in set partitions
Walaa Asakly, Noor Kezil
TL;DR
The paper studies the statistic sume on set partitions of [n], defined as the total sum of all elements preceding records across blocks in the canonical sequential form. It develops an ordinary generating-function framework, introducing P_{k,a}(x,q) and P_k(x,q) with explicit product forms, and derives exact expressions for the total sume over partitions with exactly k blocks: sum_{pi in P_{n,k}} sume(pi) = S_{n,k} sum_{a=1}^{k} a(a-1)/2 + sum_{i=1}^{k-1} ((k-i) i (i+1)/2 sum_{j=1}^{n-k} S_{n-j,k} i^{j-1}). It then aggregates to the total over all partitions: sum_{pi in P_n} sume(pi) = (1/3) B_{n+3} - (1/4) B_{n+2} - ((1/2)n+13/12) B_{n+1} - (1/12 + (1/2)n) B_n, and derives the asymptotic estimate sum ~ B_n n^3 / r^3 (1 + r/n) with r solving r e^r = n+1.
Abstract
In this paper, we aim to derive an explicit formula for the total number of elements preceding records over all set partitions of $[n]$ with exactly $k$ blocks, as well as an asymptotic estimate for the total sum of elements preceding records in all set partitions of $[n]$, expressed in terms of Bell numbers. To achieve this, we analyze the generating function that enumerates set partitions of $[n]$ according to this statistic, which we denote by $\sumelements$.