4,208 research outputs found
The government’s benefits cuts mean that families are finding it even harder to make ends meet
Rising inflation and the freezing and cutting of benefits for those in work mean that households are more squeezed than they were only a year ago – in fact wages need to have risen by 24 per cent for a family to reach the same standard of living as in 2010. Chris Goulden writes that this pressure on family budgets is likely to continue, and may even get worse
Stanley's character polynomials and coloured factorisations in the symmetric group
In Stanley [R.P. Stanley, Irreducible symmetric group characters of rectangular shape, Sém. Lothar. Combin. 50 (2003) B50d, 11 p.] the author introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. In [R.P. Stanley, A conjectured combinatorial interpretation of the normalised irreducible character values of the symmetric group, math.CO/0606467, 2006] the same author gives a conjectured combinatorial interpretation for the coefficients of the polynomials. Here, we prove the conjecture for the terms of highest degree
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California forest die-off linked to multi-year deep soil drying in 2012-2015 drought
Tree-like properties of cycle factorizations
We provide a bijection between the set of factorizations, that is, ordered
(n-1)-tuples of transpositions in whose product is (12...n),
and labelled trees on vertices. We prove a refinement of a theorem of
D\'{e}nes that establishes new tree-like properties of factorizations. In
particular, we show that a certain class of transpositions of a factorization
correspond naturally under our bijection to leaf edges of a tree. Moreover, we
give a generalization of this fact.Comment: 10 pages, 3 figure
Monotone Hurwitz numbers in genus zero
Hurwitz numbers count branched covers of the Riemann sphere with specified
ramification data, or equivalently, transitive permutation factorizations in
the symmetric group with specified cycle types. Monotone Hurwitz numbers count
a restricted subset of the branched covers counted by the Hurwitz numbers, and
have arisen in recent work on the the asymptotic expansion of the
Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study
of monotone Hurwitz numbers. We prove two results that are reminiscent of those
for classical Hurwitz numbers. The first is the monotone join-cut equation, a
partial differential equation with initial conditions that characterizes the
generating function for monotone Hurwitz numbers in arbitrary genus. The second
is our main result, in which we give an explicit formula for monotone Hurwitz
numbers in genus zero.Comment: 22 pages, submitted to the Canadian Journal of Mathematic
Annular noncrossing permutations and minimal transitive factorizations
We give two combinatorial proofs of Goulden and Jackson's formula for the
number of minimal transitive factorizations of a permutation when the
permutation has two cycles. We use the recent result of Goulden, Nica, and
Oancea on the number of maximal chains of annular noncrossing partitions of
type .Comment: 13 pages, 3 Figure
An explicit form for Kerov's character polynomials
Kerov considered the normalized characters of irreducible representations of
the symmetric group, evaluated on a cycle, as a polynomial in free cumulants.
Biane has proved that this polynomial has integer coefficients, and made
various conjectures. Recently, Sniady has proved Biane's conjectured explicit
form for the first family of nontrivial terms in this polynomial. In this
paper, we give an explicit expression for all terms in Kerov's character
polynomials. Our method is through Lagrange inversion.Comment: 17 pages, 1 figur
The number of ramified coverings of the sphere by the double torus, and a general form for higher genera
An explicit expression is obtained for the generating series for the number
of ramified coverings of the sphere by the double torus, with elementary branch
points and prescribed ramification type over infinity. Thus we are able to
prove a conjecture of Graber and Pandharipande, giving a linear recurrence
equation for the number of these coverings with no ramification over infinity.
The general form of the series is conjectured for the number of these coverings
by a surface of arbitrary genus that is at least two.Comment: 14pp.; revised version has two additional results in Section
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