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Counting Simsun Permutations by Descents
Chak-On Chow1 and Wai Chee Shiu2
1Department of Mathematics and Information Technology, Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong
cchow@alum.mit.edu
2Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
wcshiu@hkbu.edu.hk
Annals of Combinatorics 15 (4) pp.625-635 December, 2011
AMS Subject Classification: 05A15; 05A19, 05A05, 05E05, 05E10
Abstract:
We count in the present work simsun permutations of length n by their number of descents. Properties studied include the recurrence relation and real-rootedness of the generating function of the number of n-simsun permutations with k descents. By means of generating function arguments, we show that the descent number is equidistributed over n-simsun permutations and n-André permutations. We also compute the mean and variance of the random variable Xn taking values the descent number of random n-simsun permutations, and deduce that the distribution of descents over random simsun permutations of length n satisfies a central and a local limit theorem as n→+∞.
Keywords: simsun permutations, descents, Andr´e trees, asymptotically normal

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