Multi-statistic enumeration of two-stack sortable permutations

Using Zeilberger's factorization of two-stack-sortable permutations, we write a functional equation — of a strange sort — that defines their generating function according to five statistics: length, number of descents, number of right-to-left and left-to-right maxima, and a fifth statistic that is closely linked to the factorization. Then, we show how one can translate this functional equation into a polynomial one. We thus prove that our five-variable generating function for two-stack-sortable permutations is algebraic of degree 20.

[1]  W. T. Tutte,et al.  On the Enumeration of Rooted Non-Separable Planar Maps , 1964, Canadian Journal of Mathematics.

[2]  Ira M. Gessel,et al.  A combinatorial proof of the multivariable lagrange inversion formula , 1987, J. Comb. Theory, Ser. A.

[3]  T. Mansour,et al.  Involutions avoiding the class of permutations in Sk with prefix 12 , 2007 .

[4]  Robert Cori,et al.  Description Trees for Some Families of Planar Maps , 1997 .

[5]  Donald E. Knuth.  The art of computer programming: fundamental algorithms , 1969 .

[6]  Doron Zeilberger,et al.  A proof of Julian West's conjecture that the number of two-stacksortable permutations of length n is 2(3n)!/((n + 1)!(2n + 1)!) , 1992, Discret. Math..

[7]  W. T. Tutte.  A Census of Planar Maps , 1963, Canadian Journal of Mathematics.

[8]  Julian West,et al.  Raney Paths and a Combinatorial Relationship between Rooted Nonseparable Planar Maps and Two-Stack-Sortable Permutations , 1996, J. Comb. Theory, Ser. A.

[9]  Donald E. Knuth,et al.  The art of computer programming: V.1.: Fundamental algorithms , 1997 .

[10]  Julian West,et al.  Sorting Twice Through a Stack , 1993, Theor. Comput. Sci..

[11]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[12]  W. G. Brown,et al.  Enumeration of Non-Separable Planar Maps , 1963, Canadian Journal of Mathematics.

[13]  Olivier Guibert.  Combinatoire des permutations à motifs exclus en liaison avec mots, cartes planaires et tableaux de Young , 1995 .

[14]  W. G. Brown.  On the existence of square roots in certain rings of power series , 1965 .

[15]  Serge Dulucq,et al.  A combinatorial proof of J. West's conjecture , 1998, Discret. Math..

[16]  W. G. Brown,et al.  Enumeration of Quadrangular Dissections of the Disk , 1965, Canadian Journal of Mathematics.

[17]  Zhi-Cheng Gao.  The number of rooted triangular maps on a surface , 1991, J. Comb. Theory, Ser. B.

[18]  Bruno Salvy,et al.  GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable , 1994, TOMS.

[19]  Serge Dulucq,et al.  Permutations with forbidden subsequences and nonseparable planar maps , 1996, Discret. Math..

[20]  Mireille Bousquet-Mélou,et al.  A method for the enumeration of various classes of column-convex polygons , 1996, Discret. Math..