|
|
A361911
|
|
Number of set partitions of {1..n} with block-medians summing to an integer.
|
|
6
|
|
|
1, 1, 3, 10, 30, 107, 479, 2249, 11173, 60144, 351086
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
|
|
LINKS
|
|
|
EXAMPLE
|
The a(1) = 1 through a(4) = 10 set partitions:
{{1}} {{1}{2}} {{123}} {{1}{234}}
{{13}{2}} {{12}{34}}
{{1}{2}{3}} {{123}{4}}
{{124}{3}}
{{13}{24}}
{{134}{2}}
{{14}{23}}
{{1}{24}{3}}
{{13}{2}{4}}
{{1}{2}{3}{4}}
The set partition {{1,4},{2,3}} has medians {5/2,5/2}, with sum 5, so is counted under a(4).
|
|
MATHEMATICA
|
sps[{}]:={{}}; sps[set:{i_, ___}] := Join@@Function[s, Prepend[#, s]& /@ sps[Complement[set, s]]] /@ Cases[Subsets[set], {i, ___}];
Table[Length[Select[sps[Range[n]], IntegerQ[Total[Median/@#]]&]], {n, 10}]
|
|
CROSSREFS
|
For median instead of sum we have A361864.
For mean instead of median we have A361866.
A308037 counts set partitions with integer average block-size.
|
|
KEYWORD
|
nonn,more
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|