22 research outputs found
A Bijective Census of Nonseparable Planar Maps
AbstractBijections are obtained between nonseparable planar maps and two different kinds of trees: description trees and skew ternary trees. A combinatorial relation between the latter and ternary trees allows bijective enumeration and random generation of nonseparable planar maps. The involved bijections take account of the usual combinatorial parameters and give a bijective proof of formulae established by Brown and Tutte. These results, combined with a bijection due to Goulden and West, give a purely combinatorial enumeration of two-stack-sortable permutations
Restricted non-separable planar maps and some pattern avoiding permutations
Tutte founded the theory of enumeration of planar maps in a series of papers
in the 1960s. Rooted non-separable planar maps are in bijection with
West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori,
Jacquard and Schaeffer in 1997, as well as a family of permutations defined by
the avoidance of two four letter patterns. In this paper we give upper and
lower bounds on the number of multiple-edge-free rooted non-separable planar
maps. We also use the bijection between rooted non-separable planar maps and a
certain class of permutations, found by Claesson, Kitaev and Steingrimsson in
2009, to show that the number of 2-faces (excluding the root-face) in a map
equals the number of occurrences of a certain mesh pattern in the permutations.
We further show that this number is also the number of nodes in the
corresponding beta(1,0)-tree that are single children with maximum label.
Finally, we give asymptotics for some of our enumerative results.Comment: 18 pages, 14 figure
Enumeration of Stack-Sorting Preimages via a Decomposition Lemma
We give three applications of a recently-proven "Decomposition Lemma," which
allows one to count preimages of certain sets of permutations under West's
stack-sorting map . We first enumerate the permutation class
, finding a new example
of an unbalanced Wilf equivalence. This result is equivalent to the enumeration
of permutations sortable by , where is the bubble
sort map. We then prove that the sets ,
,
and are
counted by the so-called "Boolean-Catalan numbers," settling a conjecture of
the current author and another conjecture of Hossain. This completes the
enumerations of all sets of the form
for
with the exception of the set
. We also find an explicit formula for
, where
is the set of permutations in with descents.
This allows us to prove a conjectured identity involving Catalan numbers and
order ideals in Young's lattice.Comment: 20 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1903.0913
Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps
Solving the first nonmonotonic, longer-than-three instance of a classic
enumeration problem, we obtain the generating function of all
1342-avoiding permutations of length as well as an {\em exact} formula for
their number . While achieving this, we bijectively prove that the
number of indecomposable 1342-avoiding permutations of length equals that
of labeled plane trees of a certain type on vertices recently enumerated by
Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number
of rooted bicubic maps enumerated by Tutte in 1963. Moreover, turns out
to be algebraic, proving the first nonmonotonic, longer-than-three instance of
a conjecture of Zeilberger and Noonan. We also prove that
converges to 8, so in particular,