OFFSET
0,3
COMMENTS
A multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.
A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part). This is a multiset generalization of singleton-reduced phylogenetic trees (A000311).
EXAMPLE
The a(0) = 1 through a(3) = 9 trees:
() (1) (11) (111)
(12) (112)
(123)
((1)(11))
((1)(12))
((1)(23))
((2)(11))
((2)(13))
((3)(12))
The a(4) = 69 trees, with singleton leaves (x) replaced by just x:
(1111) (1112) (1122) (1123) (1234)
(1(111)) (1(112)) (1(122)) (1(123)) (1(234))
(11(11)) (11(12)) (11(22)) (11(23)) (12(34))
((11)(11)) (12(11)) (12(12)) (12(13)) (13(24))
(1(1(11))) (2(111)) (2(112)) (13(12)) (14(23))
((11)(12)) (22(11)) (2(113)) (2(134))
(1(1(12))) ((11)(22)) (23(11)) (23(14))
(1(2(11))) (1(1(22))) (3(112)) (24(13))
(2(1(11))) ((12)(12)) ((11)(23)) (3(124))
(1(2(12))) (1(1(23))) (34(12))
(2(1(12))) ((12)(13)) (4(123))
(2(2(11))) (1(2(13))) ((12)(34))
(1(3(12))) (1(2(34)))
(2(1(13))) ((13)(24))
(2(3(11))) (1(3(24)))
(3(1(12))) ((14)(23))
(3(2(11))) (1(4(23)))
(2(1(34)))
(2(3(14)))
(2(4(13)))
(3(1(24)))
(3(2(14)))
(3(4(12)))
(4(1(23)))
(4(2(13)))
(4(3(12)))
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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
mtot[m_]:=Prepend[Join@@Table[Tuples[mtot/@p], {p, Select[mps[m], Length[#]>1&&Length[#]<Length[m]&]}], m];
Table[Sum[Length[mtot[s]], {s, strnorm[n]}], {n, 0, 5}]
CROSSREFS
The case with all atoms different is A000311.
The case with all atoms equal is A196545.
The orderless version is A316652.
The unlabeled version is A330470.
The case where the leaves are sets is A330628.
The version for just normal (not strongly normal) is A330654.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 23 2019
STATUS
approved