|
|
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
|
|
|
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
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|