100 research outputs found
Additive triples of bijections, or the toroidal semiqueens problem
We prove an asymptotic for the number of additive triples of bijections
, that is, the number of pairs of
bijections such that
the pointwise sum is also a bijection. This problem is equivalent
to counting the number of orthomorphisms or complete mappings of
, to counting the number of arrangements of
mutually nonattacking semiqueens on an toroidal chessboard, and to
counting the number of transversals in a cyclic Latin square. The method of
proof is a version of the Hardy--Littlewood circle method from analytic number
theory, adapted to the group .Comment: 22 page
The action of S_n on the cohomology of M_{0,n}(R)
In recent work (math/0507514) by Etingof, Henriques, Kamnitzer, and the
author, a presentation and explicit basis was given for the rational cohomology
of the real locus \bar{M_{0,n}}(\RR) of the moduli space of stable genus 0
curves with marked points. We determine the graded character of the action
of on this space (induced by permutations of the marked points), both in
the form of a plethystic formula for the cycle index, and as an explicit
product formula for the value of the character on a given cycle type.Comment: 17 pages AMSLaTe
An integration of Euler's pentagonal partition
A recurrent formula is presented, for the enumeration of the compositions of
positive integers as sums over multisets of positive integers, that closely
resembles Euler's recurrence based on the pentagonal numbers, but where the
coefficients result from a discrete integration of Euler's coefficients. Both a
bijective proof and one based on generating functions show the equivalence of
the subject recurrences.Comment: 22 pages, 2 figures. The recurrence investigated in this paper is
essentially that proposed in Exercise 5.2.3 of Igor Pak's "Partition
bijections, a survey", Ramanujan J. 12 (2006), but casted in a different form
and, perhaps more interestingly, endowed with a bijective proof which arises
from a construction by induction on maximal part
S-Restricted Compositions Revisited
An S-restricted composition of a positive integer n is an ordered partition
of n where each summand is drawn from a given subset S of positive integers.
There are various problems regarding such compositions which have received
attention in recent years. This paper is an attempt at finding a closed- form
formula for the number of S-restricted compositions of n. To do so, we reduce
the problem to finding solutions to corresponding so-called interpreters which
are linear homogeneous recurrence relations with constant coefficients. Then,
we reduce interpreters to Diophantine equations. Such equations are not in
general solvable. Thus, we restrict our attention to those S-restricted
composition problems whose interpreters have a small number of coefficients,
thereby leading to solvable Diophantine equations. The formalism developed is
then used to study the integer sequences related to some well-known cases of
the S-restricted composition problem
The action of S_n on the cohomology of M_(0,n)(R)
In recent work by Etingof, Henriques, Kamnitzer, and the author, a presentation and explicit basis was given for the rational cohomology of the real locus M_(0,n)(R) of the moduli space of stable genus 0 curves with n marked points. We determine the graded character of the action of S_n on this space (induced by permutations of the marked points), both in the form of a plethystic formula for the cycle index, and as an explicit product formula for the value of the character on a given cycle type
- …