In this paper, we focus on the internal structural characteristics of permutation tableaux and their correspondence with linked partitions. We begin by introducing new statistics for permutation tableaux, designed to thoroughly describe various positional relationships among the topmost 1s and the rightmost restricted 0s. Subsequently, we develop two marking algorithms for permutation tableaux, each from the perspective of columns and rows. Additionally, we introduce tugging and rebound transformations, which elucidate the generative relationship from original partitions to linked partitions. As a result, we demonstrate that the construction of these two marking algorithms in permutation tableaux provides a straightforward method for enumerating the crossing number and nesting number of the corresponding linked partitions.

In this paper, we focus on the internal structural characteristics of permutation tableaux and their correspondence with linked partitions. We begin by introducing new statistics or permutation tableaux, designed to thoroughly describe various positional relationships among the topmost 1’s and the rightmost restricted 0’s. Subsequently, we develop two marking algorithms for permutation tableaux, each from the perspective of columns and rows. Additionally, we introduce tugging and rebound transformations, which elucidate the generative relationship from original partitions to linked partitions. As a result,we demonstrate that the construction of these two marking algorithms in permutation tableaux provides a straightforward method for enumerating the crossing number and nesting number of the corresponding linked partitions.

In this paper, we introduce a class of restricted symmetric permutations, called half-exceeded symmetric permutations. We deduce the enumerative formula of the permutations of { 1 , 2 , … , 2 n } and give it a refinement according to the distribution of the inverse pairs. As a consequence, we obtain new combinatorial interpretations of some well-known sequences, such as Stirling numbers of the second kind and ordered Bell numbers. Moreover, we introduce the ordered Stirling number of the second kind and establish a combinatorial proof of the recursive relation of the sequence.