Decompositions and statistics for β (1, 0)-trees and nonseparable permutations

Anders Claesson*, Sergey Kitaev, Einar Steingrímsson

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length n that can be sorted by passing it twice through a stack (where the letters on the stack have to be in increasing order) was conjectured by West, and later proved by Zeilberger. Goulden and West found a bijection from such permutations to nonseparable planar maps, and later, Jacquard and Schaeffer presented a bijection from these planar maps to certain labeled plane trees, called β (1, 0)-tree. Using generating trees, Dulucq, Gire and West showed that nonseparable planar maps are equinumerous with permutations avoiding the (classical) pattern 2413 and the barred pattern 41 over(3, ̄) 52; they called these permutations nonseparable. We give a new bijection between β (1, 0)-tree and permutations avoiding the dashed patterns 3-1-4-2 and 2-41-3. These permutations can be seen to be exactly the reverse of nonseparable permutations. Our bijection is built using decompositions of the permutations and the trees, and it translates seven statistics on the trees into statistics on the permutations. Among the statistics involved are ascents, left-to-right minima and right-to-left maxima for the permutations, and leaves and the rightmost and leftmost paths for the trees. In connection with this we give a nontrivial involution on the β (1, 0)-tree, which specializes to an involution on unlabeled rooted plane trees, where it yields interesting results. Lastly, we conjecture the existence of a bijection between nonseparable permutations and two-stack sortable permutations preserving at least four permutation statistics.

Original languageEnglish
Pages (from-to)313-328
Number of pages16
JournalAdvances in Applied Mathematics
Issue number3
Publication statusPublished - Mar 2009

Bibliographical note

Funding Information:
✩ The research presented here was supported by grant no. 060005012 from the Icelandic Research Fund.

Other keywords

  • Bijection
  • Involution
  • Nonseparable
  • Pattern avoidance
  • Planar maps
  • Stack sorting
  • Trees


Dive into the research topics of 'Decompositions and statistics for β (1, 0)-trees and nonseparable permutations'. Together they form a unique fingerprint.

Cite this