Hook Inequalities

  • Igor PakEmail author
  • Fedor Petrov
  • Viacheslav Sokolov
In enumerative combinatorics, results are usually easy to state. Essentially, you are counting the number of certain combinatorial objects: exactly, asymptotically, bijectively, or otherwise. Judging the importance of the results is also relatively easy: the more natural or interesting the objects, and the stronger or more elegant the final formula, the better. In fact, the story or the context behind the results is usually superfluous, since they speak for themselves. In the words of Gian-Carlo Rota, one of the founding fathers of modern enumerative combinatorics:

Combinatorics is an honest subject ... You count balls in a box, and you either have the right number or you haven’t. [21].

It is only occasionally that context makes the difference; this paper is an exception.

The subject of this paper is certain new combinatorial inequalities for hook numbers of Young diagrams, rooted trees, and their generalizations. In two special cases, these inequalities are known, have technical...



We are grateful to Joshua Swanson for telling us about Swanson (2018) and for his careful reading of the manuscript. The first author is thankful to Alejandro Morales and Greta Panova for numerous interesting conversations about the Naruse hook-length formula. We are also grateful to BIRS, in Banff, Canada, for hosting the first two authors at the Asymptotic Algebraic Combinatorics workshop in March 2019, where this paper was finalized. This collaboration began when the first author asked a question on MathOverflow. (See The first author was partially supported by the NSF. The second and third authors were partially supported by the RSCF (Grant 17-71-20153).


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.Steklov Mathematical InstituteSaint PetersburgRussia

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