On Whitney numbers of the order ideals of generalized fences and crowns

We solve some recurrences given by E. Munarini and N. Zagaglia Salvi proving explicit formulas for Whitney numbers of the distributive lattices of order ideals of the fence poset and crown poset. Moreover, we get a method to obtain explicit formulas for Whitney numbers of lattices of order ideals of fences with higher asymmetric peaks.

[1]  D. Kleitman,et al.  Proof techniques in the theory of finite sets , 1978 .

[2]  George E. Andrews,et al.  Special Functions: Partitions , 1999 .

[3]  Norma Zagaglia Salvi,et al.  On the rank polynomial of the lattice of order ideals of fences and crowns , 2002, Discret. Math..

[4]  N. J. Fine,et al.  Basic Hypergeometric Series and Applications , 1988 .

[5]  R. Stanley.  Log‐Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry a , 1989 .

[6]  S. G. Hoggar.  Chromatic polynomials and logarithmic concavity , 1974 .

[7]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[8]  Chris D. Godsil,et al.  A permutation group determined by an ordered set , 2003, Discret. Math..

[9]  Robert A. Proctor,et al.  Solution of Two Difficult Combinatorial Problems with Linear Algebra , 1982 .

[10]  David G. Wagner.  Logarithmic Concavity and SI2(C) , 2001, J. Comb. Theory, Ser. A.

[11]  Christian Krattenthaler,et al.  HYP and HYPQ Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively q-binomial sums and basic hypergeometric series , 1993 .

[12]  Christian Krattenthaler,et al.  HYP and HYPQ , 1995, J. Symb. Comput..

[13]  Mizan Rahman,et al.  Basic Hypergeometric Series , 1990 .

[14]  I. Anderson.  Combinatorics of Finite Sets , 1987 .

[15]  George Steiner,et al.  On estimating the number of order ideals in partial orders, with some applications , 1993 .

[16]  George Steiner,et al.  An algorithm to generate the ideals of a partial order , 1986 .