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A323955 Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} with no block containing k cyclically successive vertices, n >= 1, 2 <= k <= n + 1.

%I A323955 #5 Feb 10 2019 23:02:10
%S A323955 1,1,2,1,4,5,4,10,14,15,11,36,46,51,52,41,145,184,196,202,203,162,631,
%T A323955 806,855,869,876,877,715,3015,3847,4059,4115,4131,4139,4140,3425,
%U A323955 15563,19805,20813,21056,21119,21137,21146,21147,17722,86144,109339,114469
%N A323955 Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} with no block containing k cyclically successive vertices, n >= 1, 2 <= k <= n + 1.
%C A323955 Cyclically successive means 1 is a successor of n.
%e A323955 Triangle begins:
%e A323955     1
%e A323955     1    2
%e A323955     1    4    5
%e A323955     4   10   14   15
%e A323955    11   36   46   51   52
%e A323955    41  145  184  196  202  203
%e A323955   162  631  806  855  869  876  877
%e A323955   715 3015 3847 4059 4115 4131 4139 4140
%e A323955 Row 4 counts the following partitions:
%e A323955   {{13}{24}}      {{12}{34}}      {{1}{234}}      {{1234}}
%e A323955   {{1}{24}{3}}    {{13}{24}}      {{12}{34}}      {{1}{234}}
%e A323955   {{13}{2}{4}}    {{14}{23}}      {{123}{4}}      {{12}{34}}
%e A323955   {{1}{2}{3}{4}}  {{1}{2}{34}}    {{124}{3}}      {{123}{4}}
%e A323955                   {{1}{23}{4}}    {{13}{24}}      {{124}{3}}
%e A323955                   {{12}{3}{4}}    {{134}{2}}      {{13}{24}}
%e A323955                   {{1}{24}{3}}    {{14}{23}}      {{134}{2}}
%e A323955                   {{13}{2}{4}}    {{1}{2}{34}}    {{14}{23}}
%e A323955                   {{14}{2}{3}}    {{1}{23}{4}}    {{1}{2}{34}}
%e A323955                   {{1}{2}{3}{4}}  {{12}{3}{4}}    {{1}{23}{4}}
%e A323955                                   {{1}{24}{3}}    {{12}{3}{4}}
%e A323955                                   {{13}{2}{4}}    {{1}{24}{3}}
%e A323955                                   {{14}{2}{3}}    {{13}{2}{4}}
%e A323955                                   {{1}{2}{3}{4}}  {{14}{2}{3}}
%e A323955                                                   {{1}{2}{3}{4}}
%t A323955 spsu[_,{}]:={{}};spsu[foo_,set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,___}];
%t A323955 Table[Length[spsu[Select[Subsets[Range[n]],Select[Partition[Range[n],k,1,1],Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&],Range[n]]],{n,7},{k,2,n+1}]
%Y A323955 First column (k = 2) is A000296. Second column (k = 3) is A323949. Rightmost terms are A000110. Second to rightmost terms are A058692.
%Y A323955 Cf. A000126, A000325, A001610, A001644, A169985, A306351, A306357, A323950, A323954.
%K A323955 nonn,tabl
%O A323955 1,3
%A A323955 _Gus Wiseman_, Feb 10 2019