|
|
A320172
|
|
Number of series-reduced balanced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.
|
|
4
|
|
|
1, 2, 5, 9, 19, 38, 79, 163, 352, 750, 1633, 3558, 7783, 17020, 37338, 81920, 180399, 398600, 885101, 1975638, 4435741, 10013855, 22726109, 51807432, 118545425, 272024659, 625488420, 1440067761, 3317675261, 7644488052, 17610215982, 40547552277, 93298838972, 214516498359, 492844378878
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root. In an identity tree, all branches directly under any given node are different.
|
|
LINKS
|
|
|
EXAMPLE
|
The a(1) = 1 through a(5) = 19 rooted identity trees:
(1) (2) (3) (4) (5)
(11) (21) (22) (32)
(111) (31) (41)
((1)(2)) (211) (221)
((1)(11)) (1111) (311)
((1)(3)) (2111)
((1)(21)) (11111)
((2)(11)) ((1)(4))
((1)(111)) ((2)(3))
((1)(31))
((1)(22))
((2)(21))
((3)(11))
((1)(211))
((11)(21))
((2)(111))
((1)(1111))
((11)(111))
((1)(2)(11))
|
|
MATHEMATICA
|
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gig[m_]:=Prepend[Join@@Table[Union[Sort/@Select[Sort/@Tuples[gig/@mtn], UnsameQ@@#&]], {mtn, Select[mps[m], Length[#]>1&]}], m];
Table[Sum[Length[Select[gig[y], SameQ@@Length/@Position[#, _Integer]&]], {y, Sort /@IntegerPartitions[n]}], {n, 8}]
|
|
PROG
|
(PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
seq(n)={my(u=vector(n, n, numbpart(n)), v=vector(n)); while(u, v+=u; u=WeighT(u)-u); v} \\ Andrew Howroyd, Oct 25 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|