Discovering Hook Length Formulas by an Expansion Technique

We introduce a hook length expansion technique and explain how to discover old and new hook length formulas for partitions and plane trees. The new hook length formulas for trees obtained by our method can be proved rather easily, whereas those for partitions are much more difficult and some of them still remain open conjectures. We also develop a Maple package HookExp for computing the hook length expansion. The paper can be seen as a collection of hook length formulas for partitons and plane trees. All examples are illustrated by HookExp and, for many easy cases, expained by well-known combinatorial arguments.

[1]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[2]  William Y. C. Chen,et al.  On Postnikov's hook length formula for binary trees , 2008, Eur. J. Comb..

[3]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[4]  P. McGinn.  Missed opportunities. , 2000, Hospitals & health networks.

[5]  Jeffrey B. Remmel,et al.  A Bijective Proof of the Hook Formula for the Number of Column Strict Tableaux with Bounded Entries , 1983, Eur. J. Comb..

[6]  Frank Garvan.  A q -product Tutorial for a q -series Maple Package , 1998, WWW 1998.

[7]  Guo-Niu Han,et al.  An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths , 2008, 0804.1849.

[8]  Roland Bacher,et al.  Hooks and powers of parts in partitions , 2001 .

[9]  R. Stanley.  Some combinatorial properties of hook lengths, contents, and parts of partitions , 2008, 0807.0383.

[10]  Philippe Flajolet,et al.  Varieties of Increasing Trees , 1992, CAAP.

[11]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[12]  B. M. Fulk.  MATH , 1992 .

[13]  Emil Grosswald,et al.  The Theory of Partitions , 1984 .

[14]  Guo-Niu Han,et al.  Yet Another Generalization of Postnikov's Hook Length Formula for Binary Trees , 2008, SIAM J. Discret. Math..

[15]  Dennis Stanton,et al.  CRANKS AND T -CORES , 1990 .

[16]  Andrei Okounkov,et al.  Seiberg-Witten theory and random partitions , 2003, hep-th/0306238.

[17]  Dennis Stanton,et al.  An involution for Jacobi's identity , 1989, Discret. Math..

[18]  J. Moon.  Postnikov Identities and Seo ’ s Formulas , 2006 .

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

[20]  I. G. MacDonald,et al.  Affine root systems and Dedekind'sη-function , 1971 .

[21]  D. Foata,et al.  Combinatoire et Représentation du Groupe Symétrique , 1977 .

[22]  Guo-Niu Han,et al.  New hook length formulas for binary trees , 2008, Comb..

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

[24]  Seunghyun Seo.  A Combinatorial Proof of Postnikov's Identity and a Generalized Enumeration of Labeled Trees , 2005, Electron. J. Comb..

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

[26]  Fu Liu,et al.  (k, m)-Catalan numbers and hook length polynomials for plane trees , 2007, Eur. J. Comb..

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

[28]  Guo-Niu Han,et al.  The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications , 2008, 0805.1398.

[29]  H. Prodinger,et al.  ON “ A NOTE ON THE DISTRIBUTION OF THE THREE TYPES OF NODES IN UNIFORM BINARY TREES ” BY , 1996 .

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

[31]  Christine Bessenrodt,et al.  On hooks of Young diagrams , 1998 .

[32]  Donald E. Knuth,et al.  The art of computer programming: sorting and searching (volume 3) , 1973 .

[33]  A. Lascoux.  Symmetric Functions and Combinatorial Operators on Polynomials , 2003 .

[34]  Donald E. Knuth,et al.  PERMUTATIONS, MATRICES, AND GENERALIZED YOUNG TABLEAUX , 1970 .

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

[36]  Dominique Foata,et al.  Rearrangements of the symmetric group and enumerative properties of the tangent and secant numbers , 1974 .

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

[38]  Guo-Niu Han,et al.  Arbres minimax et polynômes d'André , 2001, Adv. Appl. Math..

[39]  Ira M. Gessel,et al.  A Refinement of Cayley's Formula for Trees , 2006, Electron. J. Comb..

[40]  T. Amdeberhan.  Differential operators, shifted parts, and hook lengths , 2008, 0807.2473.

[41]  G. James,et al.  The Representation Theory of the Symmetric Group , 2009 .