|
|
A326248
|
|
Number of crossing, nesting set partitions of {1..n}.
|
|
12
|
|
|
0, 0, 0, 0, 0, 2, 28, 252, 1890, 13020, 86564
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,6
|
|
COMMENTS
|
A set partition is crossing if it has two blocks of the form {...x,y...}, {...z,t...} where x < z < y < t or z < x < t < y, and nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t.
|
|
LINKS
|
|
|
EXAMPLE
|
The a(5) = 2 set partitions:
{{1,4},{2,3,5}}
{{1,3,4},{2,5}}
|
|
MATHEMATICA
|
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
croXQ[stn_]:=MatchQ[stn, {___, {___, x_, y_, ___}, ___, {___, z_, t_, ___}, ___}/; x<z<y<t||z<x<t<y];
nesXQ[stn_]:=MatchQ[stn, {___, {___, x_, y_, ___}, ___, {___, z_, t_, ___}, ___}/; x<z<t<y||z<x<y<t];
Table[Length[Select[sps[Range[n]], nesXQ[#]&&croXQ[#]&]], {n, 0, 8}]
|
|
CROSSREFS
|
Crossing and nesting set partitions are (both) A016098.
Crossing, capturing set partitions are A326246.
Nesting, non-crossing set partitions are A122880.
|
|
KEYWORD
|
nonn,more
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|