Counting the Number of Even and Odd AUNU Permutations with Occurrences of 123- Pattern

Abstract

Permutation groups are important tools in enumerating combinatorial objects. In AUNU permutation, the first element counted as one and its length is a prime number. This paper concerned with the counting system of AUNU permutations which contain certain subsequences to obtain some new algebraic theoretic consequences. The number of even and odd of such permutations is determined and the involutions among the counted elements. Bijections are also determined between the sets of such permutations and other combinatorial objects. A discussion was presented of lattices whose maximum length chains correspond to AUNU permutations. The results in this paper complement previous work by the Authors. We also obtained generating functions  and for the number of even (respectively odd) AUNU permutations on n letters containing exactly r=1occurrence of 123 avoiding permutation.