236 research outputs found
Dihedral Sieving Phenomena
Cyclic sieving is a well-known phenomenon where certain interesting
polynomials, especially -analogues, have useful interpretations related to
actions and representations of the cyclic group. We propose a definition of
sieving for an arbitrary group and study it for the dihedral group
of order . This requires understanding the generators of the representation
ring of the dihedral group. For odd, we exhibit several instances of
dihedral sieving which involve the generalized Fibonomial coefficients,
recently studied by Amdeberhan, Chen, Moll, and Sagan. We also exhibit an
instance of dihedral sieving involving Garsia and Haiman's -Catalan
numbers.Comment: 10 page
The cyclic sieving phenomenon: a survey
The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a
2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and
f(q) be a polynomial in q with nonnegative integer coefficients. Then the
triple (X,C,f(q)) exhibits the cyclic sieving phenomenon if, for all g in C, we
have # X^g = f(w) where # denotes cardinality, X^g is the fixed point set of g,
and w is a root of unity chosen to have the same order as g. It might seem
improbable that substituting a root of unity into a polynomial with integer
coefficients would have an enumerative meaning. But many instances of the
cyclic sieving phenomenon have now been found. Furthermore, the proofs that
this phenomenon hold often involve interesting and sometimes deep results from
representation theory. We will survey the current literature on cyclic sieving,
providing the necessary background about representations, Coxeter groups, and
other algebraic aspects as needed.Comment: 48 pages, 3 figures, the sedcond version contains numerous changes
suggested by colleagues and the referee. To appear in the London Mathematical
Society Lecture Note Series. The third version has a few smaller change
Rational associahedra and noncrossing partitions
Each positive rational number x>0 can be written uniquely as x=a/(b-a) for
coprime positive integers 0<a<b. We will identify x with the pair (a,b). In
this paper we define for each positive rational x>0 a simplicial complex
\Ass(x)=\Ass(a,b) called the {\sf rational associahedron}. It is a pure
simplicial complex of dimension a-2, and its maximal faces are counted by the
{\sf rational Catalan number} \Cat(x)=\Cat(a,b):=\frac{(a+b-1)!}{a!\,b!}. The
cases (a,b)=(n,n+1) and (a,b)=(n,kn+1) recover the classical associahedron and
its "Fuss-Catalan" generalization studied by Athanasiadis-Tzanaki and
Fomin-Reading. We prove that \Ass(a,b) is shellable and give nice product
formulas for its h-vector (the {\sf rational Narayana numbers}) and f-vector
(the {\sf rational Kirkman numbers}). We define \Ass(a,b) via {\sf rational
Dyck paths}: lattice paths from (0,0) to (b,a) staying above the line y =
\frac{a}{b}x. We also use rational Dyck paths to define a rational
generalization of noncrossing perfect matchings of [2n]. In the case (a,b) =
(n, mn+1), our construction produces the noncrossing partitions of [(m+1)n] in
which each block has size m+1.Comment: 21 pages, 8 figure
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