Hook Formulas for Skew Shapes II. Combinatorial Proofs and Enumerative Applications

The Naruse hook-length formula is a recent general formula for the number of standard Young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook-lengths. In [A. H. Morales, I. Pak, and G. Panova, Hook Formulas for Skew Shapes I. $q$-Analogues and Bijections] we gave two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. In this paper we give an elementary proof of Naruse's formula based on the case of border strips. For special border strips, we obtain curious new formulas for the Euler and $q$-Euler numbers in terms of certain Dyck path summations.

[1]  Nathan Williams,et al.  Doppelgängers: Bijections of Plane Partitions , 2016, 1602.05535.

[2]  Ian P. Goulden,et al.  Planar decompositions of tableaux and Schur function determinants , 1995, Eur. J. Comb..

[3]  H. Wilf,et al.  A probabilistic proof of a formula for the number of Young tableaux of a given shape , 1979 .

[4]  V. Kreiman.  Schubert Classes in the Equivariant K-Theory and Equivariant Cohomology of the Lagrangian Grassmannian , 2006 .

[5]  Doron Zeilberger,et al.  A short hook-lengths bijection inspired by the Greene-Nijenhuis-Wilf proof , 1984, Discret. Math..

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

[7]  Robert A. Proctor.  New Symmetric Plane Partition Identities from Invariant Theory Work of De Concini and Procesi , 1990, Eur. J. Comb..

[8]  Robert A. Sulanke,et al.  The Narayana distribution , 2002 .

[9]  Gérard Viennot,et al.  Enumeration of certain young tableaux with bounded height , 1986 .

[10]  Hiroshi Naruse,et al.  Excited Young diagrams and equivariant Schubert calculus , 2007 .

[11]  Alexei Borodin,et al.  q-Distributions on boxed plane partitions , 2009, 0905.0679.

[12]  A. P. Hillman,et al.  Reverse Plane Partitions and Tableau Hook Numbers , 1976, J. Comb. Theory A.

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

[14]  H. H. Andersen,et al.  Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p : independence of p , 1994 .

[15]  Arthur L. B. Yang,et al.  Transformations of Border Strips and Schur Function Determinants , 2004 .

[16]  Bruce E. Sagan.  PROBABILISTIC PROOFS OF HOOK LENGTH FORMULAS INVOLVING TREES , 2008 .

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

[18]  I. Goulden,et al.  Combinatorial Enumeration , 2004 .

[19]  Bruce E. Sagan,et al.  A Littlewood-Richardson rule for factorial Schur functions , 1997 .

[20]  Bruce E. Sagan,et al.  Enumeration of Partitions with Hooklengths , 1982, Eur. J. Comb..

[21]  Josef Hofbauer,et al.  Communicated by the Managing Editors , 1972 .

[22]  Christian Krattenthaler,et al.  Bijective proofs of the hook formulas for the number of standard Young tableaux, ordinary and shifted , 1995, Electron. J. Comb..

[23]  Emma Yu Jin,et al.  Outside nested decompositions of skew diagrams and Schur function determinants , 2016, Eur. J. Comb..

[24]  Nathan Williams,et al.  Subwords and Plane Partitions , 2015 .

[25]  Ilse Fischer.  A bijective proof of the hook-length formula for shifted standard tableaux , 2001 .

[26]  Igor Pak,et al.  A direct bijective proof of the hook-length formula , 1997, Discret. Math. Theor. Comput. Sci..

[27]  Matjaz Konvalinka,et al.  A bijective proof of the hook-length formula for skew shapes , 2017, Electron. Notes Discret. Math..

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

[29]  Ira M. Gessel,et al.  Determinants, Paths, and Plane Partitions , 1989 .

[30]  Igor Pak,et al.  Hook length formula and geometric combinatorics. , 2001 .

[31]  Kenneth H. Rosen,et al.  Catalan Numbers , 2002 .

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

[33]  Tewodros Amdeberhan,et al.  Multi-cores, posets, and lattice paths , 2014, Adv. Appl. Math..

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

[35]  H. O. Foulkes.  Tangent and secant numbers and representations of symmetric groups , 1976, Discret. Math..

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

[37]  Sergey Fomin,et al.  Reduced Words and Plane Partitions , 1997 .

[38]  Grigori Olshanski,et al.  Shifted Schur Functions , 1996 .

[39]  yuliy baryshnikov,et al.  Enumeration formulas for young tableaux in a diagonal strip , 2007, 0709.0498.

[40]  William Graham,et al.  Excited Young diagrams, equivariant K -theory, and Schubert varieties , 2013, 1302.3009.

[41]  S. Billey,et al.  Kostant polynomials and the cohomology ring for G/B. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[42]  B. Sagan.  The Symmetric Group , 2001 .

[43]  Yuval Roichman,et al.  Standard Young Tableaux , 2015 .

[44]  Robert A. Proctor.  Shifted plane partitions of trapezoidal shape , 1983 .

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

[46]  Alain Lascoux,et al.  Ribbon Schur Functions , 1988, Eur. J. Comb..

[47]  Richard P. Stanley,et al.  Plane Partitions: Past, Present, and Future a , 1989 .

[48]  V. Kreiman.  Schubert Classes in the Equivariant K-Theory and Equivariant Cohomology of the Grassmannian , 2005 .