The parity of the number of permutation tableaux in a fixed shape

Permutation tableaux were introduced in the study of totally positive Grassmannian cells, and are connected with the steady state of asymmetric exclusion process which is an important model from statistical mechanics. The notion of linked partitions was introduced by Dykema in the study of the unsymmetrized T-transform in free probability theory. In this paper, we mainly establish the parity of the number of permutation tableaux of a fixed shape. To achieve the aim, we utilize the linked partitions as an intermediate structure. To be specific, we first introduce the structures of rainbow linked partitions and non-rainbow linked partitions, and then we establish an involution on the set of linked partitions of {1,…,n}{1,…,n} according to these two special structures. Combining the involution with a bijection between linked partitions of {1,…,n}{1,…,n} and permutation tableaux of length n given by Chen et al. [2], we obtain the desired result that the number of permutation tableaux of a fixed shape is odd. In addition, we present several interesting correspondences among rainbow linked partitions, compositions of integers, and unimodal permutations. We also have a generalized bijection between the linked partitions and permutations.