TY - JOUR

T1 - Avoiding type (1,2) or (2,1) patterns in a partition of a set

AU - Mansour, Toufik

AU - Shattuck, Mark

PY - 2012/1/1

Y1 - 2012/1/1

N2 - A partition π of the set (n) = {1,2, . . . , n} is a collection {B1, . . . , Bk} of nonempty pairwise disjoint subsets of (n) (called blocks) whose union equals (n). In this paper, we find exact formulas and/or generating functions for the number of partitions of (n) with k blocks, where k is fixed, which avoid 3-letter patterns of type x − yz or xy − z, providing generalizations in several instances. In the particular cases of 23 − 1, 22 − 1, and 32 − 1, we are only able to find recurrences and functional equations satisfied by the generating function, since in these cases there does not appear to be a simple explicit formula for it.

AB - A partition π of the set (n) = {1,2, . . . , n} is a collection {B1, . . . , Bk} of nonempty pairwise disjoint subsets of (n) (called blocks) whose union equals (n). In this paper, we find exact formulas and/or generating functions for the number of partitions of (n) with k blocks, where k is fixed, which avoid 3-letter patterns of type x − yz or xy − z, providing generalizations in several instances. In the particular cases of 23 − 1, 22 − 1, and 32 − 1, we are only able to find recurrences and functional equations satisfied by the generating function, since in these cases there does not appear to be a simple explicit formula for it.

U2 - 10.1515/integers-2012-0004

DO - 10.1515/integers-2012-0004

M3 - מאמר

SN - 1553-1732

VL - 12

JO - Integers

JF - Integers

M1 - #A20

ER -