2,680 research outputs found
An explicit derivation of the Mobius function for Bruhat order
We give an explicit nonrecursive complete matching for the Hasse diagram of
the strong Bruhat order of any interval in any Coxeter group. This yields a new
derivation of the Mobius function, recovering a classical result due to Verma.Comment: 9 pages; final versio
Abacus models for parabolic quotients of affine Weyl groups
We introduce abacus diagrams that describe minimal length coset
representatives in affine Weyl groups of types B, C, and D. These abacus
diagrams use a realization of the affine Weyl group of type C due to Eriksson
to generalize a construction of James for the symmetric group. We also describe
several combinatorial models for these parabolic quotients that generalize
classical results in affine type A related to core partitions.Comment: 28 pages, To appear, Journal of Algebra. Version 2: Updated with
referee's comment
The enumeration of fully commutative affine permutations
We give a generating function for the fully commutative affine permutations
enumerated by rank and Coxeter length, extending formulas due to Stembridge and
Barcucci--Del Lungo--Pergola--Pinzani. For fixed rank, the length generating
functions have coefficients that are periodic with period dividing the rank. In
the course of proving these formulas, we obtain results that elucidate the
structure of the fully commutative affine permutations.Comment: 18 pages; final versio
Results and conjectures on simultaneous core partitions
An n-core partition is an integer partition whose Young diagram contains no
hook lengths equal to n. We consider partitions that are simultaneously a-core
and b-core for two relatively prime integers a and b. These are related to
abacus diagrams and the combinatorics of the affine symmetric group (type A).
We observe that self-conjugate simultaneous core partitions correspond to the
combinatorics of type C, and use abacus diagrams to unite the discussion of
these two sets of objects.
In particular, we prove that (2n)- and (2mn+1)-core partitions correspond
naturally to dominant alcoves in the m-Shi arrangement of type C_n,
generalizing a result of Fishel--Vazirani for type A. We also introduce a major
statistic on simultaneous n- and (n+1)-core partitions and on self-conjugate
simultaneous (2n)- and (2n+1)-core partitions that yield q-analogues of the
Coxeter-Catalan numbers of type A and type C.
We present related conjectures and open questions on the average size of a
simultaneous core partition, q-analogs of generalized Catalan numbers, and
generalizations to other Coxeter groups. We also discuss connections with the
cyclic sieving phenomenon and q,t-Catalan numbers.Comment: 17 pages; to appear in the European Journal of Combinatoric
Nanodiamond arrays on glass for quantification and fluorescence characterisation
Quantifying the variation in emission properties of fluorescent nanodiamonds
is important for developing their wide-ranging applicability. Directed
self-assembly techniques show promise for positioning nanodiamonds precisely
enabling such quantification. Here we show an approach for depositing
nanodiamonds in pre-determined arrays which are used to gather statistical
information about fluorescent lifetimes. The arrays were created via a layer of
photoresist patterned with grids of apertures using electron beam lithography
and then drop-cast with nanodiamonds. Electron microscopy revealed a 90%
average deposition yield across 3,376 populated array sites, with an average of
20 nanodiamonds per site. Confocal microscopy, optimised for nitrogen vacancy
fluorescence collection, revealed a broad distribution of fluorescent lifetimes
in agreement with literature. This method for statistically quantifying
fluorescent nanoparticles provides a step towards fabrication of hybrid
photonic devices for applications from quantum cryptography to sensing
Leading coefficients of Kazhdan--Lusztig polynomials for Deodhar elements
We show that the leading coefficient of the Kazhdan--Lusztig polynomial
known as is always either 0 or 1 when is a Deodhar
element of a finite Weyl group. The Deodhar elements have previously been
characterized using pattern avoidance by Billey--Warrington (2001) and
Billey--Jones (2007). In type , these elements are precisely the 321-hexagon
avoiding permutations. Using Deodhar's (1990) algorithm, we provide some
combinatorial criteria to determine when for such permutations
.Comment: 28 page
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