Periodic de Bruijn triangles: exact and asymptotic results

We study the distribution of the number of permutations with a given periodic up-down sequence w.r.t. the last entry, find exponential generating functions and prove asymptotic formulas for this distribution.

[1]  H. O. Foulkes,et al.  Enumeration of permutations with prescribed up-down and inversion sequences , 1976, Discret. Math..

[2]  L. Carlitz,et al.  Generating functions for a special class of permutations , 1975 .

[3]  Gérard Viennot,et al.  A Combinatorial Interpretation of the Seidel Generation of Genocchi Numbers , 1980 .

[4]  R. Stanley.  What Is Enumerative Combinatorics , 1986 .

[5]  Gérard Viennot,et al.  Permutations ayant une forme donnee , 1979, Discret. Math..

[6]  Désiré André,et al.  Sur les permutations alternées , 1881 .

[7]  A. Vainshtein,et al.  Counting real rational functions with all real critical values , 2002 .

[8]  Richard P. Stanley,et al.  Binomial Posets, Möbius Inversion, and Permutation Enumeration , 1976, J. Comb. Theory A.

[9]  Percy Alexander MacMahon,et al.  Second Memoir on the Compositions of Numbers , 1908 .

[10]  Michelle L. Wachs,et al.  Generalized quotients in Coxeter groups , 1988 .

[11]  Leonard Carlitz,et al.  Permutations with prescribed pattern , 1973 .

[12]  Vladimir I. Arnold,et al.  The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups , 1992 .

[13]  Richard Scoville,et al.  Generating Functions for Certain Types of Permutations , 1975, J. Comb. Theory, Ser. A.

[14]  Vladimir I. Arnold,et al.  Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics , 1991 .

[15]  Oscar Nierstrasz,et al.  A combinatorial application of matrix Riccati equations and their q-analogue , 1981, Discret. Math..

[16]  Gordon Simons,et al.  Asymptotic sine laws arising from alternating random permutations and sequences , 1996, Random Struct. Algorithms.

[17]  L. Carlitz,et al.  Enumeration of rises and falls by position , 1973, Discret. Math..

[18]  M. Kreĭn,et al.  Linear operators leaving invariant a cone in a Banach space , 1950 .

[19]  D. Dumont.  Further Triangles of Seidel-Arnold Type and Continued Fractions Related to Euler and Springer Numbers , 1995 .

[20]  de Ng Dick Bruijn.  Permutations with given ups and downs , 1970 .