14,621 research outputs found
Permutations Containing and Avoiding 123 and 132 Patterns
We prove that the number of permutations which avoid 132-patterns and have
exactly one 123-pattern equals (n-2)2^(n-3). We then give a bijection onto the
set of permutations which avoid 123-patterns and have exactly one 132-pattern.
Finally, we show that the number of permutations which contain exactly one
123-pattern and exactly one 132-pattern is (n-3)(n-4)2^(n-5).Comment: 5 page
Finitely labeled generating trees and restricted permutations
Generating trees are a useful technique in the enumeration of various
combinatorial objects, particularly restricted permutations. Quite often the
generating tree for the set of permutations avoiding a set of patterns requires
infinitely many labels. Sometimes, however, this generating tree needs only
finitely many labels. We characterize the finite sets of patterns for which
this phenomenon occurs. We also present an algorithm - in fact, a special case
of an algorithm of Zeilberger - that is guaranteed to find such a generating
tree if it exists.Comment: Accepted by J. Symb. Comp.; 12 page
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
Combinatorics of simple marked mesh patterns in 132-avoiding permutations
We present some combinatorial interpretations for coefficients appearing in
series partitioning the permutations avoiding 132 along marked mesh patterns.
We identify for patterns in which only one parameter is non zero the
combinatorial family in bijection with 132-avoiding permutations and also
preserving the statistic counted by the marked mesh pattern.Comment: 11 pages, 6 figures, submitted at FPSAC 201
- …