Hook formulas for skew shapes I. q-analogues and bijections

Abstract The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse's formula, by using factorial Schur functions and a generalization of the Hillman–Grassl correspondence , respectively. The main new results are two different q -analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. We establish explicit bijections between these objects and families of integer arrays with certain nonzero entries, which also proves the second formula.

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

[2]  Emden R. Gansner,et al.  The Hillman-Grassl Correspondence and the Enumeration of Reverse Plane Partitions , 1981, J. Comb. Theory, Ser. A.

[3]  Ezra Miller,et al.  Gröbner geometry of vertex decompositions and of flagged tableaux , 2005, math/0502144.

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

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

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

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

[8]  Pavel Galashin.  A Littlewood-Richardson rule for dual stable Grothendieck polynomials , 2017, J. Comb. Theory, Ser. A.

[9]  Richard P. Stanley,et al.  A Formula for the Specialization of Skew Schur Functions , 2016 .

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

[11]  Mark Huber,et al.  Fast perfect sampling from linear extensions , 2006, Discret. Math..

[12]  J. Remmel.  Bijective proofs of formulae for the number of standard Yound tableaux , 1982 .

[13]  Jason Bandlow,et al.  An Elementary Proof of the Hook Formula , 2008, Electron. J. Comb..

[14]  Anatoly M. Vershik.  Hook formula and related identities , 1992 .

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

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

[17]  Terence Tao,et al.  Puzzles and (equivariant) cohomology of Grassmannians , 2001, math/0112150.

[18]  John R. Stembridge,et al.  On the Fully Commutative Elements of Coxeter Groups , 1996 .

[19]  B Kostant,et al.  The nil Hecke ring and cohomology of G/P for a Kac-Moody group G. , 1986, Proceedings of the National Academy of Sciences of the United States of America.

[20]  Hariharan Narayanan.  On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients , 2006 .

[21]  Hiroshi Naruse,et al.  K-theoretic analogues of factorial Schur P- and Q-functions , 2011, 1112.5223.

[22]  G. Viennot,et al.  Une forme geometrique de la correspondance de Robinson-Schensted , 1977 .

[23]  Greta Panova,et al.  Asymptotics of the number of standard Young tableaux of skew shape , 2016, Eur. J. Comb..

[24]  Allen Knutson.  A Schubert calculus recurrence from the noncomplex W-action on G/B , 2003 .

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

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

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

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

[29]  Richard P. Stanley,et al.  On the enumeration of skew Young tableaux , 2001, Adv. Appl. Math..

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

[31]  C. Krattenthaler,et al.  Plane partitions in the work of Richard Stanley and his school , 2015, 1503.05934.

[32]  R. Stanley.  Theory and Application of Plane Partitions. Part 2 , 1971 .

[33]  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.

[34]  Greta Panova,et al.  Hook Formulas for Skew Shapes II. Combinatorial Proofs and Enumerative Applications , 2016, SIAM J. Discret. Math..

[35]  Julianna S. Tymoczko,et al.  Billey's formula in combinatorics, geometry, and topology , 2013, 1309.0254.

[36]  Christian Krattenthaler.  An Involution Principle-Free Bijective Proof of Stanley's Hook-Content Formula , 1998, Discret. Math. Theor. Comput. Sci..

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

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

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

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

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

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

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

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

[45]  Emden R. Gansner,et al.  MATRIX CORRESPONDENCES OF PLANE PARTITIONS , 1981 .

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

[47]  B. Kostant,et al.  The Nil Hecke Ring and Cohomology of , 2003 .

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

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

[50]  Christian Krattenthaler.  Another Involution Principle-Free Bijective Proof of Stanley's Hook-Content Formula , 1999, J. Comb. Theory, Ser. A.

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

[52]  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 .

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

[54]  Greta Panova,et al.  Hook formulas for skew shapes III. Multivariate and product formulas , 2017, Algebraic Combinatorics.

[55]  Igor Pak,et al.  The weighted hook length formula , 2010, J. Comb. Theory, Ser. A.

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

[57]  T. Inui,et al.  The Symmetric Group , 1990 .

[58]  Richard P. Stanley,et al.  The Conjugate Trace and Trace of a Plane Partition , 1973, J. Comb. Theory, Ser. A.

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

[60]  Greta Panova,et al.  Hook formulas for skew shapes , 2015 .

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

[62]  Alexander Yong,et al.  Equivariant Schubert calculus and jeu de taquin , 2012 .