22,796 research outputs found
Refining overpartitions by properties of non-overlined parts
We study new classes of overpartitions of numbers based on the properties of non-overlined parts. Several combinatorial identities are established by means of generating functions and bijective proofs. We show that our enumeration function satisfies a pair of infinite Ramanujantype congruences modulo 3. Lastly, by conditioning on the overlined parts of overpartitions,we give a seemingly new identity between the number of overpartitions and a certain class of ordinary partition functions. A bijective proof for this theorem also includes a partial answer to a previous request for a bijection on partitions doubly restricted by divisibility and frequency
Combinatorially interpreting generalized Stirling numbers
Let be a word in alphabet with 's and 's.
Interpreting "" as multiplication by , and "" as differentiation with
respect to , the identity , valid
for any smooth function , defines a sequence , the terms of
which we refer to as the {\em Stirling numbers (of the second kind)} of .
The nomenclature comes from the fact that when , we have , the ordinary Stirling number of the second kind.
Explicit expressions for, and identities satisfied by, the have been
obtained by numerous authors, and combinatorial interpretations have been
presented. Here we provide a new combinatorial interpretation that retains the
spirit of the familiar interpretation of as a count of
partitions. Specifically, we associate to each a quasi-threshold graph
, and we show that enumerates partitions of the vertex set of
into classes that do not span an edge of . We also discuss some
relatives of, and consequences of, our interpretation, including -analogs
and bijections between families of labelled forests and sets of restricted
partitions.Comment: To appear in Eur. J. Combin., doi:10.1016/j.ejc.2014.07.00
-Torsion Points In Finite Abelian Groups And Combinatorial Identities
The main aim of this article is to compute all the moments of the number of
-torsion elements in some type of nite abelian groups. The averages
involved in these moments are those de ned for the Cohen-Lenstra heuristics for
class groups and their adaptation for Tate-Shafarevich groups. In particular,
we prove that the heuristic model for Tate-Shafarevich groups is compatible
with the recent conjecture of Poonen and Rains about the moments of the orders
of -Selmer groups of elliptic curves. For our purpose, we are led to de ne
certain polynomials indexed by integer partitions and to study them in a
combinatorial way. Moreover, from our probabilistic model, we derive
combinatorial identities, some of which appearing to be new, the others being
related to the theory of symmetric functions. In some sense, our method
therefore gives for these identities a somehow natural algebraic context.Comment: 24 page
Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes
We employ the -spin tautological relations to provide a particular
combinatorial identity. We show that this identity is a statement equivalent to
Faber's formula for proportionalities of kappa-classes on ,
. We then prove several cases of the combinatorial identity, providing
a new proof of Faber's formula for those cases
A Probabilistic Proof of the Rogers Ramanujan Identities
The asymptotic probability theory of conjugacy classes of the finite general
linear and unitary groups leads to a probability measure on the set of all
partitions of natural numbers. A simple method of understanding these measures
in terms of Markov chains is given and compared with work on the uniform
measure. Elementary probabilistic proofs of the Rogers-Ramanujan identities
follow. As a corollary, the main case of Bailey's lemma is interpreted as
finding eigenvectors of the transition matrix of the Markov chain. It is shown
that the viewpoint of Markov chains extends to quivers.Comment: Final version, to appear in Bull LMS. The one math change is to fix a
typo in the limit in Corollary 2. We also make two historical correction
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