50 research outputs found
On the number of n-dimensional representations of SU(3), the Bernoulli numbers, and the Witten zeta function
We derive new results about properties of the Witten zeta function associated
with the group SU(3), and use them to prove an asymptotic formula for the
number of n-dimensional representations of SU(3) counted up to equivalence. Our
analysis also relates the Witten zeta function of SU(3) to a summation identity
for Bernoulli numbers discovered in 2008 by Agoh and Dilcher. We give a new
proof of that identity and show that it is a special case of a stronger
identity involving the Eisenstein series.Comment: To appear in Acta Arithmetic
Generating functions of bipartite maps on orientable surfaces
We compute, for each genus , the generating function of (labelled) bipartite maps on the orientable surface of
genus , with control on all face degrees. We exhibit an explicit change of
variables such that for each , is a rational function in the new
variables, computable by an explicit recursion on the genus. The same holds for
the generating function of rooted bipartite maps. The form of the result
is strikingly similar to the Goulden/Jackson/Vakil and
Goulden/Guay-Paquet/Novak formulas for the generating functions of classical
and monotone Hurwitz numbers respectively, which suggests stronger links
between these models. Our result complements recent results of Kazarian and
Zograf, who studied the case where the number of faces is bounded, in the
equivalent formalism of dessins d'enfants. Our proofs borrow some ideas from
Eynard's "topological recursion" that he applied in particular to even-faced
maps (unconventionally called "bipartite maps" in his work). However, the
present paper requires no previous knowledge of this topic and comes with
elementary (complex-analysis-free) proofs written in the perspective of formal
power series.Comment: 31 pages, 2 figure
Generic and special constructions of pure O-sequences
It is shown that the h-vectors of Stanley-Reisner rings of three classes of
matroids are pure O-sequences. The classes are (a) matroids that are
truncations of other matroids, or more generally of Cohen-Macaulay complexes,
(b) matroids whose dual is (rank + 2)-partite, and (c) matroids of
Cohen-Macaulay type at most five. Consequences for the computational search for
a counterexample to a conjecture of Stanley are discussed.Comment: 16 pages, v2: various small improvements, accepted by Bulletin of the
London Math. Societ
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The life and work of Major Percy Alexander MacMahon
This thesis describes the life and work of the mathematician Major Percy Alexander MacMahon (1854 - 1929). His early life as a soldier in the Royal Artillery and events which led to him embarking on a career in mathematical research and teaching are dealt with in the first two chapters. Succeeding chapters explain the work in invariant theory and partition theory which brought him to the attention of the British mathematical community and eventually resulted in a Fellowship of the Royal Society, the presidency of the London Mathematical Society, and the award of three prestigious mathematical medals and four honorary doctorates. The development and importance of his recreational mathematical work is traced and discussed. MacMahon's career in the Civil Service as Deputy Warden of the Standards at the Board of Trade is also described. Throughout the thesis, his involvement with the British Association for the Advancement of Science and other scientific organisations is highlighted. The thesis also examines possible reasons why MacMahon's work, held in very high regard at the time, did not lead to the lasting fame accorded to some of his contemporaries. Details of his personal and social life are included to give a picture of MacMahon as a real person working hard to succeed in a difficult context
Pinnacle sets of signed permutations
Pinnacle sets record the values of the local maxima for a given family of
permutations. They were introduced by Davis-Nelson-Petersen-Tenner as a dual
concept to that of peaks, previously defined by Billey-Burdzy-Sagan. In recent
years pinnacles and admissible pinnacles sets for the type symmetric group
have been widely studied. In this article we define the pinnacle set of signed
permutations of types and . We give a closed formula for the number of
type / admissible pinnacle sets and answer several other related
enumerative questions.Comment: 15 pages, 3 figures, to appear in Discrete Mathematic