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

Let P_n 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 NCP_n. Two patterns τ and υ are belong to the same NC-Wilf class if and only if the number of noncrossing set partitions in NCP_n that avoid τ is the same as the number of noncrossing set partitions in NCP_n that avoid υ, for all n≥0, that is,

#NCP_n(τ)=#NCP_n(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 NCP_n 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)
1234	1	2	5	13	31	66	127	225	373	586	881	1277
1223, 1233, 1123	1	2	5	13	33	81	193	449	1025	2305	5121	11265
1231	1	2	5	13	33	82	202	497	1224	3017	7439	18343
1213, 1232, 1221, 1122	1	2	5	13	34	89	233	610	1597	4181	10946	28657
1222, 1112	1	2	5	13	35	96	267	750	2123	6046	17303	49721
1121, 1211	1	2	5	13	35	97	275	794	2327	6905	20705	62642
1111	1	2	5	13	36	104	309	939	2905	9118	28964	92940

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)
12345	1	2	5	14	41	116	302	715	1549	3106	5831	10352
12234, 12334, 12344, 11234	1	2	5	14	41	119	334	902	2351	5945	14660	35408
12341	1	2	5	14	41	119	336	927	2527	6870	18717	51155
12233, 11233, 11223	1	2	5	14	41	121	354	1021	2901	8130	22513	61713
12343, 12134,12324	1	2	5	14	41	121	355	1032	2973	8496	24111	68017
12331, 12231	1	2	5	14	41	121	355	1033	2986	8594	24674	70757
12342, 12314	1	2	5	14	41	121	356	1044	3057	8948	26192	76674
12332, 12333, 12133, 11123, 12213, 12321, 12223, 11232	1	2	5	14	41	122	365	1094	3281	9842	29525	88574
11122, 12221, 11222	1	2	5	14	41	123	374	1147	3538	10958	34042	105997
11231, 12311	1	2	5	14	41	122	367	1117	3441	10720	33727	107012
12113, 12322, 12232, 11213	1	2	5	14	41	123	375	1157	3603	11304	35683	113219
12211, 11221	1	2	5	14	41	123	375	1158	3615	11393	36209	115940
12131	1	2	5	14	41	123	376	1168	3678	11716	37688	122261
12222, 11112	1	2	5	14	41	124	384	1210	3865	12482	40677	133572
11121, 11211, 12111	1	2	5	14	41	124	385	1221	3939	12886	42648	142544
11111	1	2	5	14	41	125	393	1265	4147	13798	46476	158170

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)
123456	1	2	5	14	42	131	407	1205	3313	8398	19691	43022
122345, 123345, 123445, 123455,112345	1	2	5	14	42	131	411	1263	3750	10721	29571	79009
123451	1	2	5	14	42	131	411	1266	3799	11152	32313	93228
123344, 112344, 112234, 122334, 122344, 112334	1	2	5	14	42	131	414	1302	4035	12274	36626	107331
123245, 123454, 123435, 121345	1	2	5	14	42	131	414	1304	4063	12497	37960	114013
123441, 123341, 122341	1	2	5	14	42	131	414	1304	4065	12530	38265	116103
123452, 123415	1	2	5	14	42	131	414	1306	4094	12766	39695	123324
123425, 123145, 123453	1	2	5	14	42	131	415	1317	4163	13090	40957	127603
123444, 122234, 123334, 111234	1	2	5	14	42	131	415	1318	4173	13150	41242	128801
112233	1	2	5	14	42	131	416	1327	4219	13315	41637	128941
123244, 121344, 122343, 112324, 112343, 121334	1	2	5	14	42	131	416	1328	4233	13430	42362	132827
122331	1	2	5	14	42	131	416	1328	4234	13446	42509	133846
123443, 122134, 123324	1	2	5	14	42	131	416	1329	4247	13544	43071	136568
123431, 123241	1	2	5	14	42	131	416	1329	4247	13545	43088	136732
123342, 123314, 123442, 122314	1	2	5	14	42	131	416	1329	4248	13560	43217	137570
123421, 112342, 123144	1	2	5	14	42	131	416	1330	4261	13658	43782	140347
112341, 123411	1	2	5	14	42	131	415	1321	4218	13533	43704	142170
111233, 122333, 111223, 122233, 112223, 112333	1	2	5	14	42	131	417	1340	4321	13939	44916	144484
121343, 122133, 112332, 123432, 122231, 123321, 123214, 123331	1	2	5	14	42	131	417	1341	4334	14041	45542	147798
122324, 123343, 112134, 123224, 121134, 123433	1	2	5	14	42	131	417	1342	4347	14142	46155	151014
122311, 112231, 123311, 112331	1	2	5	14	42	131	417	1342	4349	14173	46434	152931
122342, 112314, 123114, 123422	1	2	5	14	42	131	417	1343	4361	14257	46883	154962
122213, 123332, 111232, 121333	1	2	5	14	42	131	418	1352	4410	14463	47605	157084
123141, 121341	1	2	5	14	42	131	417	1344	4375	14372	47614	158961
122321, 121133, 112133, 112322, 112232, 123221	1	2	5	14	42	131	418	1353	4422	14553	48145	159924
122113, 112213, 123322, 122332	1	2	5	14	42	131	418	1353	4423	14567	48263	160702
123211, 112321	1	2	5	14	42	131	418	1353	4423	14568	48279	160853
123333, 111123, 122223	1	2	5	14	42	131	418	1354	4434	14643	48688	162809
121314, 123242	1	2	5	14	42	131	418	1354	4435	14655	48776	163324
121331, 122131	1	2	5	14	42	131	418	1354	4436	14670	48908	164220
111222	1	2	5	14	42	131	419	1362	4476	14821	49338	164875
112221, 122211	1	2	5	14	42	131	419	1363	4488	14914	49921	168094
111122, 112222, 122221	1	2	5	14	42	131	419	1364	4498	14980	50280	169836
111231, 123111, 112311	1	2	5	14	42	131	418	1356	4461	14859	50036	170096
111213, 122232, 123222, 112113, 122322, 121113	1	2	5	14	42	131	419	1365	4511	15081	50901	173181
122111, 112211, 111221	1	2	5	14	42	131	419	1365	4512	15096	51032	174064
121311, 112131, 121131	1	2	5	14	42	131	419	1366	4524	15183	51532	176587
122222, 111112	1	2	5	14	42	131	420	1375	4576	15431	52603	180957
111211, 112111, 121111, 111121	1	2	5	14	42	131	420	1376	4588	15521	53144	183825
111111	1	2	5	14	42	131	421	1385	4642	15795	54418	189454

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

#NCPn(τ)=#NCPn(υ).

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

#NCP_n(τ)=#NCP_n(υ).