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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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 07 2018
EXTENSIONS
Terms a(13) and beyond from Andrew Howroyd, Oct 25 2018
STATUS
approved