Half-exceeded Symmetric Permutations and the Inverse Pairs

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.