1,730 research outputs found
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
Pattern avoidance in "flattened" partitions
To flatten a set partition (with apologies to Mathematica) means to form a
permutation by erasing the dividers between its blocks. Of course, the result
depends on how the blocks are listed. For the usual listing--increasing entries
in each block and blocks arranged in increasing order of their first
entries--we count the partitions of [n] whose flattening avoids a single
3-letter pattern. Five counting sequences arise: a null sequence, the powers of
2, the Fibonacci numbers, the Catalan numbers, and the binomial transform of
the Catalan numbers.Comment: 8 page
Restricted ascent sequences and Catalan numbers
Ascent sequences are those consisting of non-negative integers in which the
size of each letter is restricted by the number of ascents preceding it and
have been shown to be equinumerous with the (2+2)-free posets of the same size.
Furthermore, connections to a variety of other combinatorial structures,
including set partitions, permutations, and certain integer matrices, have been
made. In this paper, we identify all members of the (4,4)-Wilf equivalence
class for ascent sequences corresponding to the Catalan number
C_n=\frac{1}{n+1}\binom{2n}{n}. This extends recent work concerning avoidance
of a single pattern and provides apparently new combinatorial interpretations
for C_n. In several cases, the subset of the class consisting of those members
having exactly m ascents is given by the Narayana number
N_{n,m+1}=\frac{1}{n}\binom{n}{m+1}\binom{n}{m}.Comment: 12 page
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
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