Symmetries of statistics on lattice paths between two boundaries

Abstract We prove that on the set of lattice paths with steps N = ( 0 , 1 ) and E = ( 1 , 0 ) that lie between two fixed boundaries T and B (which are themselves lattice paths), the statistics ‘number of E steps shared with B’ and ‘number of E steps shared with T’ have a symmetric joint distribution. To do so, we give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps S = ( 0 , − 1 ) at prescribed x-coordinates. We also show that a similar equidistribution result for path statistics follows from the fact that the Tutte polynomial of a matroid is independent of the order of its ground set. We extend the two theorems to k-tuples of paths between two boundaries, and we give some applications to Dyck paths, generalizing a result of Deutsch, to watermelon configurations, to pattern-avoiding permutations, and to the generalized Tamari lattice. Finally, we prove a conjecture of Nicolas about the distribution of degrees of k consecutive vertices in k-triangulations of a convex n-gon. To achieve this goal, we provide a new statistic-preserving bijection between certain k-tuples of non-crossing paths and k-flagged semistandard Young tableaux, which is based on local moves reminiscent of jeu de taquin.

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

[2]  Emeric Deutsch,et al.  An involution on Dyck paths and its consequences , 1999, Discret. Math..

[3]  Nicholas A. Loehr,et al.  Conjectured Statistics for the Higher q, t-Catalan Sequences , 2005, Electron. J. Comb..

[4]  I. Gessel,et al.  Binomial Determinants, Paths, and Hook Length Formulae , 1985 .

[5]  Carlo Vanderzande,et al.  Lattice Models of Polymers , 1998 .

[6]  Frank Sottile,et al.  Tableau Switching: Algorithms and Applications , 1996, J. Comb. Theory, Ser. A.

[7]  Philippe Flajolet,et al.  Basic analytic combinatorics of directed lattice paths , 2002, Theor. Comput. Sci..

[8]  Christian Krattenthaler.  Watermelon configurations with wall interaction: exact and asymptotic results , 2005 .

[9]  Michelle L. Wachs,et al.  Flagged Schur Functions, Schubert Polynomials, and Symmetrizing Operators , 1985, J. Comb. Theory, Ser. A.

[10]  Ira M. Gessel,et al.  Lattice Paths and Faber Polynomials , 1997 .

[11]  Tomoki Nakamigawa,et al.  A generalization of diagonal flips in a convex polygon , 2000, Theor. Comput. Sci..

[12]  Sergi Elizalde.  A bijection between 2-triangulations and pairs of non-crossing Dyck paths , 2007, J. Comb. Theory, Ser. A.

[13]  H. Crapo,et al.  The Tutte polynomial , 1969, 1707.03459.

[14]  -nilpotent -ideals in () having a fixed class of nilpotence: combinatorics and enumeration , 2002 .

[15]  Daniel J. Kleitman,et al.  Strong Versions of Sperner's Theorem , 1976, J. Comb. Theory, Ser. A.

[16]  Luis Serrano,et al.  Maximal Fillings of Moon Polyominoes, Simplicial Complexes, and Schubert Polynomials , 2010, Electron. J. Comb..

[17]  J. W. Essam,et al.  Return polynomials for non-intersecting paths above a surface on the directed square lattice , 2001 .

[18]  Jacobus H. Koolen,et al.  On Line Arrangements in the Hyperbolic Plane , 2002, Eur. J. Comb..

[19]  W. T. Tutte,et al.  A Contribution to the Theory of Chromatic Polynomials , 1954, Canadian Journal of Mathematics.

[20]  Marc Noy,et al.  Lattice path matroids: enumerative aspects and Tutte polynomials , 2003, J. Comb. Theory, Ser. A.

[21]  Carlos M. Nicolás.  Another bijection between $2$-triangulations and pairs of non-crossing Dyck paths , 2009 .

[22]  Federico Ardila.  The Catalan matroid , 2003, J. Comb. Theory, Ser. A.

[23]  Ira M. Gessel,et al.  A Factorization for Formal Laurent Series and Lattice Path Enumeration , 1980, J. Comb. Theory A.

[24]  Jakob Jonsson,et al.  Generalized triangulations and diagonal-free subsets of stack polyominoes , 2005, J. Comb. Theory, Ser. A.

[25]  Emeric Deutsch,et al.  A bijection on Dyck paths and its consequences , 1998, Discret. Math..

[26]  Mireille Bousquet-Mélou,et al.  The Number of Intervals in the m-Tamari Lattices , 2011, Electron. J. Comb..

[27]  John Irving,et al.  The number of lattice paths below a cyclically shifting boundary , 2009, J. Comb. Theory, Ser. A.

[28]  Robin J. Chapman,et al.  Simple formulas for lattice paths avoiding certain periodic staircase boundaries , 2009, J. Comb. Theory, Ser. A.

[29]  K. Humphreys.  A history and a survey of lattice path enumeration , 2010 .

[30]  F. Bergeron,et al.  Higher Trivariate Diagonal Harmonics via generalized Tamari Posets , 2011, 1105.3738.

[31]  T. V. Narayana,et al.  Lattice Path Combinatorics With Statistical Applications , 1979 .

[32]  Marc Renault,et al.  Lost (and Found) in Translation: André's Actual Method and Its Application to the Generalized Ballot Problem , 2008, Am. Math. Mon..

[33]  Michael E. Fisher,et al.  Walks, walls, wetting, and melting , 1984 .

[34]  Philippe Duchon,et al.  On the enumeration and generation of generalized Dyck words , 2000, Discret. Math..