445 research outputs found
Finitely labeled generating trees and restricted permutations
Generating trees are a useful technique in the enumeration of various
combinatorial objects, particularly restricted permutations. Quite often the
generating tree for the set of permutations avoiding a set of patterns requires
infinitely many labels. Sometimes, however, this generating tree needs only
finitely many labels. We characterize the finite sets of patterns for which
this phenomenon occurs. We also present an algorithm - in fact, a special case
of an algorithm of Zeilberger - that is guaranteed to find such a generating
tree if it exists.Comment: Accepted by J. Symb. Comp.; 12 page
Combinatorics of patience sorting piles
Despite having been introduced in 1962 by C.L. Mallows, the combinatorial
algorithm Patience Sorting is only now beginning to receive significant
attention due to such recent deep results as the Baik-Deift-Johansson Theorem
that connect it to fields including Probabilistic Combinatorics and Random
Matrix Theory.
The aim of this work is to develop some of the more basic combinatorics of
the Patience Sorting Algorithm. In particular, we exploit the similarities
between Patience Sorting and the Schensted Insertion Algorithm in order to do
things that include defining an analog of the Knuth relations and extending
Patience Sorting to a bijection between permutations and certain pairs of set
partitions. As an application of these constructions we characterize and
enumerate the set S_n(3-\bar{1}-42) of permutations that avoid the generalized
permutation pattern 2-31 unless it is part of the generalized pattern 3-1-42.Comment: 19 pages, LaTeX; uses pstricks; view PS, not DVI; use dvips + ps2pdf,
not dvi2pdf; part of FPSAC'05 proceedings; v3: final journal version, revised
Section 3.
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,
The enumeration of permutations avoiding 2143 and 4231
We enumerate the pattern class Av(2143, 4231) and completely describe its permutations. The main tools are simple permutations and monotone grid classes
On the diagram of 132-avoiding permutations
The diagram of a 132-avoiding permutation can easily be characterized: it is
simply the diagram of a partition. Based on this fact, we present a new
bijection between 132-avoiding and 321-avoiding permutations. We will show that
this bijection translates the correspondences between these permutations and
Dyck paths given by Krattenthaler and by Billey-Jockusch-Stanley, respectively,
to each other. Moreover, the diagram approach yields simple proofs for some
enumerative results concerning forbidden patterns in 132-avoiding permutations.Comment: 20 pages; additional reference is adde
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