49 research outputs found
Cyclic sieving and cluster multicomplexes
Reiner, Stanton, and White \cite{RSWCSP} proved results regarding the
enumeration of polygon dissections up to rotational symmetry. Eu and Fu
\cite{EuFu} generalized these results to Cartan-Killing types other than A by
means of actions of deformed Coxeter elements on cluster complexes of Fomin and
Zelevinsky \cite{FZY}. The Reiner-Stanton-White and Eu-Fu results were proven
using direct counting arguments. We give representation theoretic proofs of
closely related results using the notion of noncrossing and semi-noncrossing
tableaux due to Pylyavskyy \cite{PN} as well as some geometric realizations of
finite type cluster algebras due to Fomin and Zelevinsky \cite{FZClusterII}.Comment: To appear in Adv. Appl. Mat
Cyclic Sieving of Increasing Tableaux and small Schr\"oder Paths
An increasing tableau is a semistandard tableau with strictly increasing rows
and columns. It is well known that the Catalan numbers enumerate both
rectangular standard Young tableaux of two rows and also Dyck paths. We
generalize this to a bijection between rectangular 2-row increasing tableaux
and small Schr\"oder paths. We demonstrate relations between the jeu de taquin
for increasing tableaux developed by H. Thomas and A. Yong and the
combinatorics of tropical frieze patterns. We then use this jeu de taquin to
present new instances of the cyclic sieving phenomenon of V. Reiner, D.
Stanton, and D. White, generalizing results of D. White and of J. Stembridge.Comment: 20 page
Combinatorics for Certain Skew Young Tableaux, Dyck Paths, Triangulations, and Dissections
We present combinatorial bijections and identities between certain skew Young
tableaux, Dyck paths, triangulations, and dissections.Comment: 22 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
Minimal Permutations and 2-Regular Skew Tableaux
Bouvel and Pergola introduced the notion of minimal permutations in the study
of the whole genome duplication-random loss model for genome rearrangements.
Let denote the set of minimal permutations of length
with descents, and let . They derived that
and , where is the -th
Catalan number. Mansour and Yan proved that . In
this paper, we consider the problem of counting minimal permutations in
with a prescribed set of ascents. We show that such
structures are in one-to-one correspondence with a class of skew Young
tableaux, which we call -regular skew tableaux. Using the determinantal
formula for the number of skew Young tableaux of a given shape, we find an
explicit formula for . Furthermore, by using the Knuth equivalence,
we give a combinatorial interpretation of a formula for a refinement of the
number .Comment: 19 page