|
|
A326249
|
|
Number of capturing set partitions of {1..n} that are not nesting.
|
|
9
|
|
|
0, 0, 0, 0, 0, 1, 9, 55, 283, 1324, 5838, 24744
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,7
|
|
COMMENTS
|
Capturing is a weaker condition than nesting. A set partition is capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t, and nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t. For example, {{1,3,5},{2,4}} is capturing but not nesting, so is counted under a(5).
|
|
LINKS
|
|
|
EXAMPLE
|
The a(6) = 9 set partitions:
{{1},{2,4,6},{3,5}}
{{1,3,5},{2,4},{6}}
{{1,3,6},{2,4},{5}}
{{1,3,6},{2,5},{4}}
{{1,4,6},{2},{3,5}}
{{1,4,6},{2,5},{3}}
{{1,3,5},{2,4,6}}
{{1,2,4,6},{3,5}}
{{1,3,5,6},{2,4}}
|
|
MATHEMATICA
|
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
capXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<t<y||z<x<y<t];
nesXQ[stn_]:=MatchQ[stn, {___, {___, x_, y_, ___}, ___, {___, z_, t_, ___}, ___}/; x<z<t<y||z<x<y<t];
Table[Length[Select[sps[Range[n]], !nesXQ[#]&&capXQ[#]&]], {n, 0, 8}]
|
|
CROSSREFS
|
MM-numbers of capturing, non-nesting multiset partitions are A326260.
Nesting set partitions are A016098.
Capturing set partitions are A326243.
Non-crossing, nesting set partitions are A122880 (conjectured).
|
|
KEYWORD
|
nonn,more
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|