Skip to main content
SpringerLink
Log in

Counting Dyck Paths by Area and Rank

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

The set of Dyck paths of length 2n inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution of two statistics for Dyck paths: area (the area under the path) and rank (the rank in the lattice). While area for Dyck paths has been studied, pairing it with this rank function seems new, and we get an interesting (q, t)-refinement of the Catalan numbers. We present two decompositions of the corresponding generating function: One refines an identity of Carlitz and Riordan; the other refines the notion of γ-nonnegativity, and is based on a decomposition of the lattice of noncrossing partitions due to Simion and Ullman. Further, Biane’s correspondence and a result of Stump allow us to conclude that the joint distribution of area and rank for Dyck paths equals the joint distribution of length and reflection length for the permutations lying below the n-cycle (12· · ·n) in the absolute order on the symmetric group.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andrews G.E.: An introduction to Ramanujan’s “lost" notebook. Amer. Math. Monthly 86(2), 89–108 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andrews G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  3. Armstrong, D.: Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Amer. Math. Soc. 202(949) (2009)

  4. Armstrong D., Stump C., Thomas H.: A uniform bijection between nonnesting and noncrossing partitions. Trans. Amer. Math. Soc. 365(8), 4121–4151 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bandlow, J., Killpatrick, K.: An area-to-inv bijection between Dyck paths and 312- avoiding permutations. Electron. J. Combin. 8(1), #R40 (2001)

  6. Biane P.: Some properties of crossings and partitions. Discrete Math. 175(1-3), 41–53 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bilotta S. et al.: Catalan structures and catalan pairs. Theoret. Comput. Sci. 502(2), 239–248 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Brändén, P.: Sign-graded posets, unimodality of W-polynomials and the Charney-Davis conjecture. Electron. J. Combin. 11(2), #R9 (2004)

  9. Brändén P., Claesson A., Steingrímsson E.: Catalan continued fractions and increasing subsequences in permutations. Discrete Math. 258(1-3), 275–287 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Carlitz, L.: Problem: q-analog of the lagrange expansion. In: Abstracts and Problems from the Conference on Eulerian Series and Applications. Pennsylvania State University, Pennsylvania (1974)

  11. Carlitz L., Riordan J.: Two element lattice permutation numbers and their q-generalization. Duke Math. J. 31(3), 371–388 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  12. Edelman P.H.: On inversions and cycles in permutations. European J. Combin. 8(3), 269–279 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ferrari L., Pinzani R.: Lattices of lattice paths. J. Statist. Plann. Inference 135(1), 77–92 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Flajolet P.: Combinatorial aspects of continued fractions. Discrete Math. 32(2), 125–161 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  15. Foata, D., Schützenberger, M.-P.: Théorie Géométrique des Polynômes Eulériens. Lecture Notes in Math., Vol. 138. Springer-Verlag, Berlin (1970)

  16. Gal Ś.R.: Real root conjecture fails for five- and higher-dimensional spheres. Discrete Comput. Geom. 34(2), 269–284 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Haglund, J.: The q, t-Catalan Numbers and the Space of Diagonal Harmonics. Univ. Lecture Ser., Vol. 41. American Mathematical Society, Providence, RI (2008)

  18. Hall, H.T.: Meanders in a cayley graph. arXiv:math/0606170v1 (2006)

  19. Hersh, P.: Deformation of chains via a local symmetric group action. Electron. J. Combin. 6, #R27 (1999)

  20. Nevo E., Petersen T.K.: On γ-vectors satisfying the Kruskal-Katona inequalities. Discrete Comput. Geom. 45(3), 503–521 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Nevo E., Petersen T.K., Tenner B.E.: The γ-vector of a barycentric subdivision. J. Combin. Theory Ser. A 118(4), 1364–1380 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. Odlyzko A.M., Wilf H.S.: The editor’s corner: n coins in a fountain. Amer. Math. Monthly 95(9), 840–843 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  23. Petersen T.K.: On the shard intersection order of a Coxeter group. SIAMJ. Discrete Math. 27(4), 1880–1912 (2013)

    Article  MATH  Google Scholar 

  24. Petersen T.K.: The sorting index. Adv. Appl. Math. 47(3), 615–630 (2011)

    Article  MATH  Google Scholar 

  25. Postnikov A., Reiner V., Williams L.: Faces of generalized permutohedra. Doc. Math. 13, 207–273 (2008)

    MATH  MathSciNet  Google Scholar 

  26. Reiner V.: Non-crossing partitions for classical reflection groups. Discrete Math. 177(1-3), 195–222 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sapounakis, A., Tasoulas, I., Tsikouras, P.: On the dominance partial ordering of Dyck paths. J. Integer Seq. 9(2), Article 06.2.5 (2006)

  28. Shareshian J., Wachs M.L.: Eulerian quasisymmetric functions. Adv. Math. 225(6), 2921–2966 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  29. Simion R., Ullman D.: On the structure of the lattice of noncrossing partitions. Discrete Math. 98(3), 193–206 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  30. Stanley, R.P.: Enumerative Combinatorics, Vol. 2. Cambridge Stud. Adv. Math., Vol. 62. Cambridge University Press, Cambridge (1999)

  31. Stembridge J.R.: Coxeter cones and their h-vectors. Adv. Math. 217(5), 1935–1961 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. Stump C.: More bijective catalan combinatorics on permutations and on signed permutations. J. Combin. 4(4), 419–447 (2013)

    Article  MathSciNet  Google Scholar 

  33. Zeng, J.: An expansion formula for the inversions and excedances in the symmetric group. arXiv:1206.3510 (2012)

Download references

Author information

Authors and Affiliations

  1. School of Informatics and Computing, Indiana University, Bloomington, IN, 47408, USA

    Saúl A. Blanco

  2. Department of Mathematical Sciences, DePaul University, Chicago, IL, 60614, USA

    T. Kyle Petersen

Authors
  1. Saúl A. Blanco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Kyle Petersen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saúl A. Blanco.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Blanco, S.A., Petersen, T.K. Counting Dyck Paths by Area and Rank. Ann. Comb. 18, 171–197 (2014). https://doi.org/10.1007/s00026-014-0218-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-014-0218-9

Mathematics Subject Classification

Keywords

Search

Navigation