Counting Linear Extensions of Posets with Determinants of Hook Lengths

We introduce a class of posets, which includes both ribbon posets (skew shapes) and $d$-complete posets, such that their number of linear extensions is given by a determinant of a matrix whose entries are products of hook lengths. We also give $q$-analogues of this determinantal formula in terms of the major index and inversion statistics. As applications, we give families of tree posets whose numbers of linear extensions are given by generalizations of Euler numbers, we draw relations to Naruse-Okada's positive formulas for the number of linear extensions of skew $d$-complete posets, and we give polynomiality results analogous to those of descent polynomials by Billey-Burdzy-Sagan and Diaz-Lopez et. al.

[1]  Robert A. Proctor,et al.  Minuscule Elements of Weyl Groups, the Numbers Game, andd-Complete Posets , 1999 .

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

[3]  Igor Pak,et al.  Counting Linear Extensions of Restricted Posets , 2018, Electron. J. Comb..

[4]  Michelle L. Wachs,et al.  q-Hook length formulas for forests , 1989, J. Comb. Theory, Ser. A.

[5]  Jang Soo Kim,et al.  Hook length property of d-complete posets via q-integrals , 2019, J. Comb. Theory, Ser. A.

[6]  (q,t)-Deformations of multivariate hook product formulae , 2009, 0909.0086.

[7]  Bruce E. Sagan,et al.  Descent polynomials , 2017, Discret. Math..

[8]  A. C. Aitken.  XXVI.—The Monomial Expansion of Determinantal Symmetric Functions , 1943 .

[9]  S. Okada,et al.  Skew hook formula for $d$-complete posets via equivariant $K$-theory , 2019, Algebraic Combinatorics.

[10]  Robert A. Proctor.  Dynkin Diagram Classification of λ-Minuscule Bruhat Lattices and of d-Complete Posets , 1999 .

[11]  Pakawut Jiradilok,et al.  Roots of Descent Polynomials and an Algebraic Inequality on Hook Lengths , 2019, Electron. J. Comb..

[12]  B. Sagan,et al.  Permutations with given peak set , 2012, 1209.0693.

[13]  G. Brightwell,et al.  Counting linear extensions , 1991 .

[14]  G. B. Mathews,et al.  Combinatory Analysis. Vol. II , 1915, The Mathematical Gazette.

[15]  J. S. Frame,et al.  The Hook Graphs of the Symmetric Group , 1954, Canadian Journal of Mathematics.

[16]  Michelle L. Wachs,et al.  Permutation statistics and linear extensions of posets , 1991, J. Comb. Theory, Ser. A.

[17]  Greta Panova,et al.  Hook formulas for skew shapes I. q-analogues and bijections , 2015, J. Comb. Theory, Ser. A.

[18]  Alexander Garver,et al.  Combinatorics of Exceptional Sequences in Type A , 2015, Electron. J. Comb..

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

[20]  Sergi Elizalde,et al.  Wilf equivalence relations for consecutive patterns , 2018, Adv. Appl. Math..

[21]  R. Thrall,et al.  A combinatorial problem. , 1952 .

[22]  Rolf H. Möhring,et al.  Computationally Tractable Classes of Ordered Sets , 1989 .

[23]  Donald E. Knuth,et al.  The Art of Computer Programming: Volume 3: Sorting and Searching , 1998 .

[24]  Mike D. Atkinson,et al.  On computing the number of linear extensions of a tree , 1990 .

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

[26]  Ira M. Gessel,et al.  Counting Permutations with Given Cycle Structure and Descent Set , 1993, J. Comb. Theory A.