13 research outputs found
Avoidance of Partitions of a Three-element Set
Klazar defined and studied a notion of pattern avoidance for set partitions,
which is an analogue of pattern avoidance for permutations. Sagan considered
partitions which avoid a single partition of three elements. We enumerate
partitions which avoid any family of partitions of a 3-element set as was done
by Simion and Schmidt for permutations. We also consider even and odd set
partitions. We provide enumerative results for set partitions restricted by
generalized set partition patterns, which are an analogue of the generalized
permutation patterns of Babson and Steingr{\'{\i}}msson. Finally, in the spirit
of work done by Babson and Steingr{'{\i}}msson, we will show how these
generalized partition patterns can be used to describe set partition
statistics.Comment: 23 pages, 2 tables, 1 figure, to appear in Advances in Applied
Mathematic
Permutation patterns and statistics
Let S_n denote the symmetric group of all permutations of the set {1, 2,
...,n} and let S = \cup_{n\ge0} S_n. If Pi is a set of permutations, then we
let Av_n(Pi) be the set of permutations in S_n which avoid every permutation of
Pi in the sense of pattern avoidance. One of the celebrated notions in pattern
theory is that of Wilf-equivalence, where Pi and Pi' are Wilf equivalent if
#Av_n(Pi)=#Av_n(Pi') for all n\ge0. In a recent paper, Sagan and Savage
proposed studying a q-analogue of this concept defined as follows. Suppose
st:S->N is a permutation statistic where N represents the nonnegative integers.
Consider the corresponding generating function, F_n^{st}(Pi;q) = sum_{sigma in
Av_n(Pi)} q^{st sigma}, and call Pi,Pi' st-Wilf equivalent if
F_n^{st}(Pi;q)=F_n^{st}(Pi';q) for all n\ge0. We present the first in-depth
study of this concept for the inv and maj statistics. In particular, we
determine all inv- and maj-Wilf equivalences for any Pi containd in S_3. This
leads us to consider various q-analogues of the Catalan numbers, Fibonacci
numbers, triangular numbers, and powers of two. Our proof techniques use
lattice paths, integer partitions, and Foata's fundamental bijection. We also
answer a question about Mahonian pairs raised in the Sagan-Savage article.Comment: 28 pages, 5 figures, tightened up the exposition, noted that some of
the conjectures have been prove
Mahonian Pairs
We introduce the notion of a Mahonian pair. Consider the set, P^*, of all
words having the positive integers as alphabet. Given finite subsets S,T of
P^*, we say that (S,T) is a Mahonian pair if the distribution of the major
index, maj, over S is the same as the distribution of the inversion number,
inv, over T. So the well-known fact that maj and inv are equidistributed over
the symmetric group, S_n, can be expressed by saying that (S_n,S_n) is a
Mahonian pair. We investigate various Mahonian pairs (S,T) with S different
from T. Our principal tool is Foata's fundamental bijection f: P^* -> P^* since
it has the property that maj w = inv f(w) for any word w. We consider various
families of words associated with Catalan and Fibonacci numbers. We show that,
when restricted to words in {1,2}^*, f transforms familiar statistics on words
into natural statistics on integer partitions such as the size of the Durfee
square. The Rogers-Ramanujan identities, the Catalan triangle, and various
q-analogues also make an appearance. We generalize the definition of Mahonian
pairs to infinite sets and use this as a tool to connect a partition bijection
of Corteel-Savage-Venkatraman with the Greene-Kleitman decomposition of a
Boolean algebra into symmetric chains. We close with comments about future work
and open problems.Comment: Minor changes suggested by the referees and updated status of the
problem of finding new Mahonian pairs; [email protected] and [email protected]
In Praise of an Elementary Identity of Euler
We survey the applications of an elementary identity used by Euler in one of
his proofs of the Pentagonal Number Theorem. Using a suitably reformulated
version of this identity that we call Euler's Telescoping Lemma, we give
alternate proofs of all the key summation theorems for terminating
Hypergeometric Series and Basic Hypergeometric Series, including the
terminating Binomial Theorem, the Chu--Vandermonde sum, the Pfaff--Saalch\" utz
sum, and their -analogues. We also give a proof of Jackson's -analog of
Dougall's sum, the sum of a terminating, balanced, very-well-poised
sum. Our proofs are conceptually the same as those obtained by the WZ method,
but done without using a computer. We survey identities for Generalized
Hypergeometric Series given by Macdonald, and prove several identities for
-analogs of Fibonacci numbers and polynomials and Pell numbers that have
appeared in combinatorial contexts. Some of these identities appear to be new.Comment: Published versio