3 research outputs found
Subword counting and the incidence algebra
The Pascal matrix, , is an upper diagonal matrix whose entries are the
binomial coefficients. In 1993 Call and Velleman demonstrated that it satisfies
the beautiful relation in which has the numbers 1, 2, 3, etc.
on its superdiagonal and zeros elsewhere. We generalize this identity to the
incidence algebras and of functions on words and
permutations, respectively. In the entries of and count
subwords; in they count permutation patterns. Inspired by
vincular permutation patterns we define what it means for a subword to be
restricted by an auxiliary index set ; this definition subsumes both factors
and (scattered) subwords. We derive a theorem for words corresponding to the
Reciprocity Theorem for patterns in permutations: Up to sign, the coefficients
in the Mahler expansion of a function counting subwords restricted by the set
is given by a function counting subwords restricted by the complementary
set
Towards m-Cambrian Lattices
For positive integers and , we introduce a family of lattices
associated to the Cambrian lattice of
the dihedral group . We show that satisfies
some basic properties of a Fuss-Catalan generalization of ,
namely that and
\bigl\lvert\mathcal{C}_{k}^{(m)}\bigr\rvert=\mbox{Cat}^{(m)}\bigl(I_{2}(k)\bigr).
Subsequently, we prove some structural and topological properties of these
lattices---namely that they are trim and EL-shellable---which were known for
before. Remarkably, our construction coincides in the case
with the -Tamari lattice of parameter 3 due to Bergeron and
Pr{\'e}ville-Ratelle. Eventually, we investigate this construction in the
context of other Coxeter groups, in particular we conjecture that the lattice
completion of the analogous construction for the symmetric group
and the long cycle is isomorphic to the
-Tamari lattice of parameter .Comment: 20 pages, 13 figures. The results of this paper are subsumed by
arXiv:1312.2520, and it will therefore not be publishe