449 research outputs found
A multiset hook length formula and some applications
A multiset hook length formula for integer partitions is established by using
combinatorial manipulation. As special cases, we rederive three hook length
formulas, two of them obtained by Nekrasov-Okounkov, the third one by Iqbal,
Nazir, Raza and Saleem, who have made use of the cyclic symmetry of the
topological vertex. A multiset hook-content formula is also proved.Comment: 19 pages; 3 figure
A Quantitative Study of Pure Parallel Processes
In this paper, we study the interleaving -- or pure merge -- operator that
most often characterizes parallelism in concurrency theory. This operator is a
principal cause of the so-called combinatorial explosion that makes very hard -
at least from the point of view of computational complexity - the analysis of
process behaviours e.g. by model-checking. The originality of our approach is
to study this combinatorial explosion phenomenon on average, relying on
advanced analytic combinatorics techniques. We study various measures that
contribute to a better understanding of the process behaviours represented as
plane rooted trees: the number of runs (corresponding to the width of the
trees), the expected total size of the trees as well as their overall shape.
Two practical outcomes of our quantitative study are also presented: (1) a
linear-time algorithm to compute the probability of a concurrent run prefix,
and (2) an efficient algorithm for uniform random sampling of concurrent runs.
These provide interesting responses to the combinatorial explosion problem
Partition into heapable sequences, heap tableaux and a multiset extension of Hammersley's process
We investigate partitioning of integer sequences into heapable subsequences
(previously defined and established by Mitzenmacher et al). We show that an
extension of patience sorting computes the decomposition into a minimal number
of heapable subsequences (MHS). We connect this parameter to an interactive
particle system, a multiset extension of Hammersley's process, and investigate
its expected value on a random permutation. In contrast with the (well studied)
case of the longest increasing subsequence, we bring experimental evidence that
the correct asymptotic scaling is . Finally
we give a heap-based extension of Young tableaux, prove a hook inequality and
an extension of the Robinson-Schensted correspondence
Algebraic aspects of increasing subsequences
We present a number of results relating partial Cauchy-Littlewood sums,
integrals over the compact classical groups, and increasing subsequences of
permutations. These include: integral formulae for the distribution of the
longest increasing subsequence of a random involution with constrained number
of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as
new proofs of old formulae; relations of these expressions to orthogonal
polynomials on the unit circle; and explicit bases for invariant spaces of the
classical groups, together with appropriate generalizations of the
straightening algorithm.Comment: LaTeX+amsmath+eepic; 52 pages. Expanded introduction, new references,
other minor change
Symmetric Group Character Degrees and Hook Numbers
In this article we prove the following result: that for any two natural
numbers k and j, and for all sufficiently large symmetric groups Sym(n), there
are k disjoint sets of j irreducible characters of Sym(n), such that each set
consists of characters with the same degree, and distinct sets have different
degrees. In particular, this resolves a conjecture most recently made by
Moret\'o. The methods employed here are based upon the duality between
irreducible characters of the symmetric groups and the partitions to which they
correspond. Consequently, the paper is combinatorial in nature.Comment: 24 pages, to appear in Proc. London Math. So
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