182 research outputs found
Pattern avoidance in compositions and multiset permutations
We study pattern avoidance by combinatorial objects other than permutations,
namely by ordered partitions of an integer and by permutations of a multiset.
In the former case we determine the generating function explicitly, for integer
compositions of n that avoid a given pattern of length 3 and we show that the
answer is the same for all such patterns. We also show that the number of
multiset permutations that avoid a given three-letter pattern is the same for
all such patterns, thereby extending and refining earlier results of Albert,
Aldred et al., and by Atkinson, Walker and Linton. Further, the number of
permutations of a multiset S, with a_i copies of i for i = 1, ..., k, that
avoid a given permutation pattern in S_3 is a symmetric function of the a_i's,
and we will give here a bijective proof of this fact first for the pattern
(123), and then for all patterns in S_3 by using a recently discovered
bijection of Amy N. Myers.Comment: 8 pages, no figur
An update on the middle levels problem
The middle levels problem is to find a Hamilton cycle in the middle levels,
M_{2k+1}, of the Hasse diagram of B_{2k+1} (the partially ordered set of
subsets of a 2k+1-element set ordered by inclusion). Previously, the best
result was that M_{2k+1} is Hamiltonian for all positive k through k=15. In
this note we announce that M_{33} and M_{35} have Hamilton cycles. The result
was achieved by an algorithmic improvement that made it possible to find a
Hamilton path in a reduced graph of complementary necklace pairs having
129,644,790 vertices, using a 64-bit personal computer.Comment: 11 pages, 5 figure
Regularly spaced subsums of integer partitions
For integer partitions , where , we study the sum of the parts of odd index. We show
that the average of this sum, over all partitions of , is of the
form More
generally, we study the sum of the parts whose
indices lie in a given arithmetic progression and we show that the average of
this sum, over all partitions of , is of the form
, with explicitly given
constants . Interestingly, for odd and we have
, so in this case the error term is of lower order. The methods used
involve asymptotic formulas for the behavior of Lambert series and the Zeta
function of Hurwitz.
We also show that if is the number of partitions of the sum of
whose parts of even index is , then for every , agrees with a
certain universal sequence, Sloane's sequence \texttt{#A000712}, for
but not for any larger
A generatingfunctionology approach to a problem of Wilf
Wilf posed the following problem: determine asymptotically as
the probability that a randomly chosen part size in a randomly chosen
composition of n has multiplicity m. One solution of this problem was given by
Hitczenko and Savage. In this paper, we study this question using the
techniques of generating functions and singularity analysis.Comment: 12 page
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