2,057 research outputs found
Counting occurrences of some subword patterns
We find generating functions the number of strings (words) containing a
specified number of occurrences of certain types of order-isomorphic classes of
substrings called subword patterns. In particular, we find generating functions
for the number of strings containing a specified number of occurrences of a
given 3-letter subword pattern.Comment: 9 page
The classification of Wada-type representations of braid groups
We give a classification of Wada-type representations of the braid groups,
and solutions of a variant of the set-theoretical Yang-Baxter equation adapted
to the free-product group structure. As a consequence, we prove Wada's
conjecture: There are only seven types of Wada-type representations up to
certain symmetries.Comment: 13 pages, 2 figures: Added Lemma 2.2, which was implicit in the
previous version without proof and more explanation
Enumerations relating braid and commutation classes
We obtain an upper and lower bound for the number of reduced words for a
permutation in terms of the number of braid classes and the number of
commutation classes of the permutation. We classify the permutations that
achieve each of these bounds, and enumerate both cases.Comment: 19 page
Embeddings between partially commutative groups: two counterexamples
In this note we give two examples of partially commutative subgroups of
partially commutative groups. Our examples are counterexamples to the Extension
Graph Conjecture and to the Weakly Chordal Conjecture of Kim and Koberda,
\cite{KK}. On the other hand we extend the class of partially commutative
groups for which it is known that the Extension Graph Conjecture holds, to
include those with commutation graph containing no induced or . In
the process, some new embeddings of surface groups into partially commutative
groups emerge.Comment: 15 pages, 5 figures; to appear in Journal of Algebr
Interval structure of the Pieri formula for Grothendieck polynomials
We give a combinatorial interpretation of a Pieri formula for double
Grothendieck polynomials in terms of an interval of the Bruhat order. Another
description had been given by Lenart and Postnikov in terms of chain
enumerations. We use Lascoux's interpretation of a product of Grothendieck
polynomials as a product of two kinds of generators of the 0-Hecke algebra, or
sorting operators. In this way we obtain a direct proof of the result of Lenart
and Postnikov and then prove that the set of permutations occuring in the
result is actually an interval of the Bruhat order.Comment: 27 page
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