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Hook Inequalities

  • Igor PakEmail author
  • Fedor Petrov
  • Viacheslav Sokolov
Article
  • 28 Downloads
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...

Notes

Acknowledgments

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 http://mathoverflow.net/q/243846). The first author was partially supported by the NSF. The second and third authors were partially supported by the RSCF (Grant 17-71-20153).

References

  1. [1] K. Barrese, N. Loehr, J. Remmel, and B. E. Sagan. \(m\)-level rook placements. J. Combin. Theory, Ser. A 124 (2014), 130–165.MathSciNetCrossRefGoogle Scholar
  2. [2] B. Beáta. Bijective proofs of the hook formula for rooted trees. Ars Combin. 106 (2012), 483–494.Google Scholar
  3. [3] E. Beckenbach and R. Bellman. Inequalities. Springer, 1961.Google Scholar
  4. [4] P. Brändén. Unimodality, log-concavity, real-rootedness and beyond. In Handbook of Enumerative Combinatorics, pp. 437–483. CRC Press, 2015.Google Scholar
  5. [5] F. Brenti. Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update. In Jerusalem Combinatorics, pp. 71–89. AMS, 1994.Google Scholar
  6. [6] G. Brightwell and P. Winkler. Counting linear extensions. Order 8 (1991), 225–247.Google Scholar
  7. [7] Yu. D. Burago and V. A. Zalgaller. Geometric Inequalities. Springer, 1988.Google Scholar
  8. [8] I. Ciocan-Fontanine, M. Konvalinka, and I. Pak. The weighted hook length formula. J. Combin. Theory, Ser. A 118 (2011), 1703–1717.MathSciNetCrossRefGoogle Scholar
  9. [9] S. Dittmer and I. Pak. Counting linear extensions of restricted posets. arXiv:1802.06312, 2018.
  10. [10] J. S. Frame, G. de B. Robinson, and R. M. Thrall. The hook graphs of the symmetric group. Canad. J. Math. 6 (1954), 316–324.MathSciNetCrossRefGoogle Scholar
  11. [11] A. Hammett and B. Pittel. How often are two permutations comparable? Trans. AMS 360 (2008), 4541–4568.MathSciNetCrossRefGoogle Scholar
  12. [12] G. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities (2nd ed.). Cambridge Univ. Press. , 1952.Google Scholar
  13. 13] B. F. Jones. Singular Chern classes of Schubert varieties via small resolution. Int. Math. Res. Not. 2010 (2010), 1371–1416.Google Scholar
  14. [14] D. Knuth. The Art of Computer Programming, Vol. III. Addison-Wesley, 1973.Google Scholar
  15. [15] A. Marshall, I. Olkin, and B. C. Arnold. Inequalities: Theory of Majorization and Its Applications (2nd ed.). Springer, 2011.Google Scholar
  16. [16] A. H. Morales, I. Pak, and G. Panova. Hook formulas for skew shapes I. \(q\)-analogues and bijections. J. Combin. Theory, Ser. A 154 (2018), 350–405.Google Scholar
  17. [17] A. H. Morales, I. Pak, and G. Panova. Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications. SIAM J. Discrete Math. 31 (2017), 1953–1989.MathSciNetCrossRefGoogle Scholar
  18. [18] A. H. Morales, I. Pak, and G. Panova. Asymptotics for the number of standard Young tableaux of skew shape. European J. Combin. 70 (2018), 26–49.MathSciNetCrossRefGoogle Scholar
  19. [19] I. Pak. Notices of the AMS Notices of the AMS 66:7 (2019), 1109–1112.Google Scholar
  20. [20] R. A. Proctor. \(d\)-Complete posets generalize Young diagrams for the hook product formula. RIMS Kôkyûroku 1913 (2014), 120–140.Google Scholar
  21. [21] G.-C. Rota and D. Sharp. Mathematics, Philosophy and Artificial Intelligence, dialogue in Los Alamos Science, No. 12 (Spring/Summer 1985), 94–104.Google Scholar
  22. [22] B. E. Sagan and Y. N. Yeh. Probabilistic algorithms for trees. Fibonacci Quart. 27 (1989), 201–208.Google Scholar
  23. [23] L. A. Shepp. The XYZ conjecture and the FKG inequality. Ann. Probab. 10 (1982), 824–827.MathSciNetCrossRefGoogle Scholar
  24. [24] R. P. Stanley, Enumerative Combinatorics, Vols. 1 (2nd ed.) and 2. Cambridge Univ. Press, 2012, 1999.Google Scholar
  25. [25] R. Sulzgruber. Symmetry properties of the Novelli–Pak–Stoyanovskii algorithm. In Proc. 26th FPSAC (Chicago, USA), DMTCS, Nancy, 2014, 205–216.Google Scholar
  26. [26] J. P. Swanson. On the existence of tableaux with given modular major index. Algebraic Combinatorics 1 (2018), 3–21.MathSciNetCrossRefGoogle Scholar

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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