# Classes of Pk, k=4,5,6, according to pattern τ avoiding noncrossing set partition and the numbers #P1(1212,τ)... #P12(1212,τ).

Let Pn be the set of all set partitions of [n]={1,2,...,n}. A set partition is said to be noncrossing if it avoids the pattern 1212. We denote the set of all noncrossing set partitions of [n] by NCPn. Two patterns τ and υ are belong to the same NC-Wilf class if and only if the number of noncrossing set partitions in NCPn that avoid τ is the same as the number of noncrossing set partitions in NCPn that avoid υ, for all n≥0, that is,

### #NCPn(τ)=#NCPn(υ).

In NCP4 there are 14 noncrossing set partitions and in NCP5 there are 42 noncrossing set partitions (it is well known that the number of noncrossing set partitions in NCPn is given by the \$n\$th Catalan number 1/(n+1)binomial(2n,n). In the table below presented the all the NC-Wilf classes of patterns of length four, five and six.

Pattern τ #NCP1(τ) #NCP2(τ) #NCP3(τ) #NCP4(τ) #NCP5(τ) #NCP6(τ) #NCP7(τ) #NCP8(τ) #NCP9(τ) #NCP10(τ) #NCP11(τ) #NCP12(τ)
12341251331661272253735868811277
1223, 1233, 112312513338119344910252305512111265
123112513338220249712243017743918343
1213, 1232, 1221, 1122125133489233610159741811094628657
1222, 1112125133596267750212360461730349721
1121, 1211125133597275794232769052070562642
11111251336104309939290591182896492940

Pattern τ #NCP1(τ) #NCP2(τ) #NCP3(τ) #NCP4(τ) #NCP5(τ) #NCP6(τ) #NCP7(τ) #NCP8(τ) #NCP9(τ) #NCP10(τ) #NCP11(τ) #NCP12(τ)
12345125144111630271515493106583110352
12234, 12334, 12344, 112341251441119334902235159451466035408
123411251441119336927252768701871751155
12233, 11233, 1122312514411213541021290181302251361713
12343, 12134,1232412514411213551032297384962411168017
12331, 1223112514411213551033298685942467470757
12342, 1231412514411213561044305789482619276674
12332, 12333, 12133, 11123, 12213, 12321, 12223, 1123212514411223651094328198422952588574
11122, 12221, 112221251441123374114735381095834042105997
11231, 123111251441122367111734411072033727107012
12113, 12322, 12232, 112131251441123375115736031130435683113219
12211, 112211251441123375115836151139336209115940
121311251441123376116836781171637688122261
12222, 111121251441124384121038651248240677133572
11121, 11211, 121111251441124385122139391288642648142544
111111251441125393126541471379846476158170

Pattern τ #NCP1(τ) #NCP2(τ) #NCP3(τ) #NCP4(τ) #NCP5(τ) #NCP6(τ) #NCP7(τ) #NCP8(τ) #NCP9(τ) #NCP10(τ) #NCP11(τ) #NCP12(τ)
12345612514421314071205331383981969143022
122345, 123345, 123445, 123455,112345125144213141112633750107212957179009
123451125144213141112663799111523231393228
123344, 112344, 112234, 122334, 122344, 1123341251442131414130240351227436626107331
123245, 123454, 123435, 1213451251442131414130440631249737960114013
123441, 123341, 1223411251442131414130440651253038265116103
123452, 1234151251442131414130640941276639695123324
123425, 123145, 1234531251442131415131741631309040957127603
123444, 122234, 123334, 1112341251442131415131841731315041242128801
1122331251442131416132742191331541637128941
123244, 121344, 122343, 112324, 112343, 1213341251442131416132842331343042362132827
1223311251442131416132842341344642509133846
123443, 122134, 1233241251442131416132942471354443071136568
123431, 1232411251442131416132942471354543088136732
123342, 123314, 123442, 1223141251442131416132942481356043217137570
123421, 112342, 1231441251442131416133042611365843782140347
112341, 1234111251442131415132142181353343704142170
111233, 122333, 111223, 122233, 112223, 1123331251442131417134043211393944916144484
121343, 122133, 112332, 123432, 122231, 123321, 123214, 1233311251442131417134143341404145542147798
122324, 123343, 112134, 123224, 121134, 1234331251442131417134243471414246155151014
122311, 112231, 123311, 1123311251442131417134243491417346434152931
122342, 112314, 123114, 1234221251442131417134343611425746883154962
122213, 123332, 111232, 1213331251442131418135244101446347605157084
123141, 1213411251442131417134443751437247614158961
122321, 121133, 112133, 112322, 112232, 1232211251442131418135344221455348145159924
122113, 112213, 123322, 1223321251442131418135344231456748263160702
123211, 1123211251442131418135344231456848279160853
123333, 111123, 1222231251442131418135444341464348688162809
121314, 1232421251442131418135444351465548776163324
121331, 1221311251442131418135444361467048908164220
1112221251442131419136244761482149338164875
112221, 1222111251442131419136344881491449921168094
111122, 112222, 1222211251442131419136444981498050280169836
111231, 123111, 1123111251442131418135644611485950036170096
111213, 122232, 123222, 112113, 122322, 1211131251442131419136545111508150901173181
122111, 112211, 1112211251442131419136545121509651032174064
121311, 112131, 1211311251442131419136645241518351532176587
122222, 1111121251442131420137545761543152603180957
111211, 112111, 121111, 1111211251442131420137645881552153144183825
1111111251442131421138546421579554418189454