The proof of a conjecture of Simion for certain partitions
A partition of an integer \(N\) is a non-increasing sequence \(\lambda=\{\lambda_1,\lambda_2,\dots ,\lambda_m\},\) with \(\lambda_1\leq n, \sum_{i=1}^{m}\lambda_i=N\), and let \(\{\lambda'\}\) be its conjugate, i.e. \(\lambda'_i=|\{j:\lambda_j\geq i\}|\), with \(\lambda_1\leq m.\) Consider the \(m\times n\) grid with the Ferrers diagram of partition \(\lambda\) removed. Furthermore denote by \(N(m,n,\lambda)\) the number of paths in the \(m\times n\) grid such that the path starts at \((0,0)\) (at the left-lower corner) and ends at \((n,m)\) (at the right-upper corner) and at each step it goes one unit up (\(+(0,1)\)) or one unit right (\(+(1,0)\)) but not inside the removed \(\lambda\). A sequence \(\{a_1,a_2,\dots, a_n\}\) is said to be unimodal if, for some \(k\), \(a_1\leq \dots \leq a_k\) and \(a_{k+1}\leq \dots\leq a_n\). The author proves a conjecture of Simion in the case when \(\lambda\) is self-conjugate. It reads: if \(\lambda\) is self-conjugate and \(t\) is a non-negative integer then the sequence \(N(\lambda'_1,\lambda_1+t,\lambda),N(\lambda'_1+1,\lambda_1+t-1,\lambda),\dots, N(\lambda'_1+t,\lambda_1,\lambda)\) is unimodal.
scientific article; zbMATH DE number 1533344