login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n.
12

%I #15 Dec 21 2020 19:33:06

%S 1,1,2,7,39,236,1836,16123,162008,1802945,22012335,291290460,

%T 4144907830,62986968311,1016584428612,17344929138791,311618472138440,

%U 5875109147135658,115894178676866576,2385755803919949337,51133201045333895149,1138659323863266945177,26296042933904490636133

%N Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n.

%C 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).

%e Non-isomorphic representatives of the a(4) = 39 trees, with singleton leaves (x) replaced by just x:

%e (1111) (1112) (1122) (1123) (1234)

%e (1(111)) (1(112)) (1(122)) (1(123)) (1(234))

%e (11(11)) (11(12)) (11(22)) (11(23)) (12(34))

%e ((11)(11)) (12(11)) (12(12)) (12(13)) ((12)(34))

%e (1(1(11))) (2(111)) ((11)(22)) (2(113)) (1(2(34)))

%e ((11)(12)) (1(1(22))) (23(11))

%e (1(1(12))) ((12)(12)) ((11)(23))

%e (1(2(11))) (1(2(12))) (1(1(23)))

%e (2(1(11))) ((12)(13))

%e (1(2(13)))

%e (2(1(13)))

%e (2(3(11)))

%o (PARI) \\ See links in A339645 for combinatorial species functions.

%o cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sEulerT(x*Ser(v[1..n])), n )); x*Ser(v)}

%o InequivalentColoringsSeq(cycleIndexSeries(15)) \\ _Andrew Howroyd_, Dec 11 2020

%Y The case with all atoms equal or all atoms different is A000669.

%Y Not requiring singleton-reduction gives A330465.

%Y Labeled versions are A316651 (normal orderless) and A330471 (strongly normal).

%Y The case where the leaves are sets is A330626.

%Y Row sums of A339645.

%Y Cf. A000311, A005121, A005804, A141268, A213427, A292504, A292505, A318812, A318848, A318849, A330467, A330469, A330474, A330624.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 22 2019

%E Terms a(7) and beyond from _Andrew Howroyd_, Dec 11 2020