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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Cyclically successive means 1 is a successor of n.

LINKS

Table of n, a(n) for n=1..49.

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.

Cf. A000126, A000325, A001610, A001644, A169985, A306351, A306357, A323950, A323954.

Sequence in context: A047949 A222986 A222906 * A197011 A326060 A321794

Adjacent sequences:  A323952 A323953 A323954 * A323956 A323957 A323958

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Feb 10 2019

STATUS

approved

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Last modified January 18 09:26 EST 2022. Contains 350454 sequences. (Running on oeis4.)