# Classes of P5 according to pattern τ avoiding partitions and the numbers #P11(τ).

A set partition of the set [n]={1,2,....,n} is a collection of disjoint blocks B_1,B_2,...,B_d whose union is [n]. We choose the ordering of the blocks so that they satisfy min B1 < min B2 <...< min Bd. We represent such a set partition by a canonical sequence π1π2...πn with πi=j if i in B_j. We say that a partition π contains a partition σ if the canonical sequence of π contains a subsequence that is order-isomorphic to the canonical sequence of σ. Two partitions σ and σ' are equivalent if there is a size-preserving bijection between σ-avoiding and σ'-avoiding partitions.

Pattern τ #P11(τ)
12314,12324,12334,12341,12342,12343,12344,12345175275
12313213423
12323,12234,12332,12123,12132,12213,12231,12312,12321,12331,12134223191
12233,12133238379
11223, 11232276670
11234282503
12131285503
11233288157
12223,12232,12322,12333,12311,12113288543
11231322218
11213323663
11123348887
12112,12122,12212,12221362447
12121364317
11212364341
12211373270
11221376556
11222378365
11122379805
11112,11121,11211,12111,12222441009
11111556711