Enumeration of Type D Permutations with Alternating Runs

  • Emily X. L. Gao
  • Brian Y. Sun


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.


Alternating runs peaks valleys up signed permutations type D permutations 

Mathematics Subject Classification

Primary 05A05 Secondary 05A15 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHenan University of Economics and LawZhengzhouPeople’s Republic of China
  2. 2.Department of Mathematics and System ScienceXinjiang UniversityUrumqiPeople’s Republic of China

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