12,560 research outputs found
Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes
We put recent results by Chen, Deng, Du, Stanley and Yan on crossings and
nestings of matchings and set partitions in the larger context of the
enumeration of fillings of Ferrers shape on which one imposes restrictions on
their increasing and decreasing chains. While Chen et al. work with
Robinson-Schensted-like insertion/deletion algorithms, we use the growth
diagram construction of Fomin to obtain our results. We extend the results by
Chen et al., which, in the language of fillings, are results about
--fillings, to arbitrary fillings. Finally, we point out that, very
likely, these results are part of a bigger picture which also includes recent
results of Jonsson on --fillings of stack polyominoes, and of results of
Backelin, West and Xin and of Bousquet-M\'elou and Steingr\'\i msson on the
enumeration of permutations and involutions with restricted patterns. In
particular, we show that our growth diagram bijections do in fact provide
alternative proofs of the results by Backelin, West and Xin and by
Bousquet-M\'elou and Steingr\'\i msson.Comment: AmS-LaTeX; 27 pages; many corrections and improvements of
short-comings; thanks to comments by Mireille Bousquet-Melou and Jakob
Jonsson, the final section is now much more profound and has additional
result
Partition Statistics Equidistributed with the Number of Hook Difference One Cells
Let be a partition, viewed as a Young diagram. We define the hook
difference of a cell of to be the difference of its leg and arm
lengths. Define to be the number of cells of with
hook difference one. In the paper of Buryak and Feigin (arXiv:1206.5640),
algebraic geometry is used to prove a generating function identity which
implies that is equidistributed with , the largest part of a
partition that appears at least twice, over the partitions of a given size. In
this paper, we propose a refinement of the theorem of Buryak and Feigin and
prove some partial results using combinatorial methods. We also obtain a new
formula for the q-Catalan numbers which naturally leads us to define a new
q,t-Catalan number with a simple combinatorial interpretation
State complexity of catenation combined with a boolean operation: a unified approach
In this paper we study the state complexity of catenation combined with
symmetric difference. First, an upper bound is computed using some combinatoric
tools. Then, this bound is shown to be tight by giving a witness for it.
Moreover, we relate this work with the study of state complexity for two other
combinations: catenation with union and catenation with intersection. And we
extract a unified approach which allows to obtain the state complexity of any
combination involving catenation and a binary boolean operation
Enumerations of Permutations Simultaneously Avoiding a Vincular and a Covincular Pattern of Length 3
Vincular and covincular patterns are generalizations of classical patterns
allowing restrictions on the indices and values of the occurrences in a
permutation. In this paper we study the integer sequences arising as the
enumerations of permutations simultaneously avoiding a vincular and a
covincular pattern, both of length 3, with at most one restriction. We see
familiar sequences, such as the Catalan and Motzkin numbers, but also some
previously unknown sequences which have close links to other combinatorial
objects such as lattice paths and integer partitions. Where possible we include
a generating function for the enumeration. One of the cases considered settles
a conjecture by Pudwell (2010) on the Wilf-equivalence of barred patterns. We
also give an alternative proof of the classic result that permutations avoiding
123 are counted by the Catalan numbers.Comment: 24 pages, 11 figures, 2 table
Frame patterns in n-cycles
In this paper, we study the distribution of the number of occurrences of the
simplest frame pattern, called the pattern, in -cycles. Given an
-cycle , we say that a pair matches the
pattern if and as we traverse around in a clockwise direction
starting at and ending at , we never encounter a with .
We say that is a nontrivial -match if .
Also, an -cycle is incontractible if there is no such that
immediately follows in .
We show that the number of incontractible -cycles in the symmetric group
is , where is the number of derangements in .
Further, we prove that the number of -cycles in with exactly
-matches can be expressed as a linear combination of binomial coefficients
of the form where . We also show that the
generating function of raised to the number of nontrivial
-matches in over all incontractible -cycles in is a new
-analogue of , which is different from the -analogues of the
derangement numbers that have been studied by Garsia and Remmel and by Wachs.
We show that there is a rather surprising connection between the charge
statistic on permutations due to Lascoux and Sch\"uzenberger and our
polynomials in that the coefficient of the smallest power of in
is the number of permutations in whose charge
path is a Dyck path. Finally, we show that and are the number of partitions
of for sufficiently large
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