Classes of P6 according to pattern τ avoiding partitions
and the numbers #Pk(τ), k=10,11,12,13,14.



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 #P10(τ) #P11(τ) #P12(τ) #P13(τ) #P14(τ)
123415,123425,123435,123445,123451,123452
123453,123454,123455,12345686472
12341489291
123413,12342489348
123134,12314389375
12324189684450407
1233148968445040823055921197896162983208
1231428968445040823055921197896162983209
123124,123145,123214,123234,123243,123245
123324,123341,123342,123345,123412,123421
123423,123431,123432,123434,123441,123442,12344389711
123144,123244,12334490135
12134292027
12231492339
12234192341
12132492369
121334,122334,121343,12234392767
121345,12234593074
12314193082
12324293084
121344,12234493135
123114,123224,123334,123343,123411,123422,123433,12344493136
121234,12213494448
12313294686
12321394712
12231395008
12133295037
123123,123312,12332195058
12323195086
12132395087
12233195434
12134195455
12131495460
122133,12123395461
11234295485
12234295486
12232495511
11232495513
112334,11234395828
11234595904
11234495909
121134,12223497377
11223497599
11233297700522415
11232397700522417
12331397828523144
12313197828523161
12313397831
12311397852
12131397872
12113297882
12131297898
121223,121232,121322,122123,122132,122213,122231
122312,122321,123112,123122,123212,123221,123223
123233,123323,123331,12333297921
12112397945
123121,123232979475248702926845
121213979475248702926847
12232397947524871
12123197948
12132197972
11234197987
11231497992
11223398023
12213198173
12211398222
12133198242
122311,123311,123211,12332298246
12233298296
121333,12233398321
121133,12223398345
11231299685
11213299730540193
11212399730540195
11213499908
11231399914
11232199990
112223,112232,11232299993
112213100010
112231100031
112331100240
112133100332
112333100338
121131101338
121311101342
121113,122223,122232,122322,123111,123222,123333101350
111223,111232101434
111234101662
111233101837
112131102778
112113102786
112311102802
111213103869
111231103905
111123104726
1212121051345874793449505
1221211051345874793449509
121221105135
122112105150
112122105176
121122105188
112212105192
122211105226
112221105233
111222105314
121222,122122,122212,122221105594
121112105614592671
112112105614592676
121121105618
1212111056365929763504916
1121211056365929763504918
1112121056365929763504921
112222105878
122111105886
111221105893
112211105895
111122105901
111112,111121,111211,112111,121111,122222107964
111111112124