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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.
3
1, 1, 2, 1, 4, 5, 4, 10, 14, 15, 11, 36, 46, 51, 52, 41, 145, 184, 196, 202, 203, 162, 631, 806, 855, 869, 876, 877, 715, 3015, 3847, 4059, 4115, 4131, 4139, 4140, 3425, 15563, 19805, 20813, 21056, 21119, 21137, 21146, 21147, 17722, 86144, 109339, 114469
OFFSET
1,3
COMMENTS
Cyclically successive means 1 is a successor of n.
EXAMPLE
Triangle begins:
1
1 2
1 4 5
4 10 14 15
11 36 46 51 52
41 145 184 196 202 203
162 631 806 855 869 876 877
715 3015 3847 4059 4115 4131 4139 4140
Row 4 counts the following partitions:
{{13}{24}} {{12}{34}} {{1}{234}} {{1234}}
{{1}{24}{3}} {{13}{24}} {{12}{34}} {{1}{234}}
{{13}{2}{4}} {{14}{23}} {{123}{4}} {{12}{34}}
{{1}{2}{3}{4}} {{1}{2}{34}} {{124}{3}} {{123}{4}}
{{1}{23}{4}} {{13}{24}} {{124}{3}}
{{12}{3}{4}} {{134}{2}} {{13}{24}}
{{1}{24}{3}} {{14}{23}} {{134}{2}}
{{13}{2}{4}} {{1}{2}{34}} {{14}{23}}
{{14}{2}{3}} {{1}{23}{4}} {{1}{2}{34}}
{{1}{2}{3}{4}} {{12}{3}{4}} {{1}{23}{4}}
{{1}{24}{3}} {{12}{3}{4}}
{{13}{2}{4}} {{1}{24}{3}}
{{14}{2}{3}} {{13}{2}{4}}
{{1}{2}{3}{4}} {{14}{2}{3}}
{{1}{2}{3}{4}}
MATHEMATICA
spsu[_, {}]:={{}}; spsu[foo_, set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsu[Select[foo, Complement[#, Complement[set, s]]=={}&], Complement[set, s]]]/@Cases[foo, {i, ___}];
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}]
CROSSREFS
First column (k = 2) is A000296. Second column (k = 3) is A323949. Rightmost terms are A000110. Second to rightmost terms are A058692.
Sequence in context: A222986 A222906 A359900 * A197011 A326060 A321794
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Feb 10 2019
STATUS
approved