Abstract
In the study of enumeration polynomials of signed permutations of rank n, which is known as a Coxeter group of type B, Chow and Ma found that alternating runs of up signed permutations are closely related to peaks and valleys of these permutations. Notice that even-singed permutations of rank n, which is also called a Coxeter group of type D, forms a subgroup of signed permutations of index 2, we study the number of type D permutations according to alternating runs and consider how alternating runs connect with peaks and valleys. We find in this paper that the generating function of alternating runs of up even-signed permutations can be expressed as those generating functions of peaks and valleys of up even-signed permutations, which partially provide an affirmative answer to a conjecture by Chow and Ma. Additionally, we establish a recurrence for the generating function of alternating runs and an identity on alternating runs of type D permutations.
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Gao, E.X.L., Sun, B.Y. Enumeration of Type D Permutations with Alternating Runs. Results Math 73, 77 (2018). https://doi.org/10.1007/s00025-018-0840-7
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DOI: https://doi.org/10.1007/s00025-018-0840-7