A368099 - OEIS

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A368099

Triangle read by rows where T(n,k) is the number of non-isomorphic k-element sets of finite nonempty multisets with cardinalities summing to n, or strict multiset partitions of weight n and length k.

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 12, 5, 1, 0, 7, 28, 22, 5, 1, 0, 11, 66, 83, 31, 5, 1, 0, 15, 134, 252, 147, 34, 5, 1, 0, 22, 280, 726, 620, 203, 35, 5, 1, 0, 30, 536, 1946, 2283, 1069, 235, 35, 5, 1, 0, 42, 1043, 4982, 7890, 5019, 1469, 248, 35, 5, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

EXAMPLE

Triangle begins:

0 1

0 2 1

0 3 4 1

0 5 12 5 1

0 7 28 22 5 1

0 11 66 83 31 5 1

0 15 134 252 147 34 5 1

0 22 280 726 620 203 35 5 1

0 30 536 1946 2283 1069 235 35 5 1

0 42 1043 4982 7890 5019 1469 248 35 5 1

...

Row n = 4 counts the following representatives:

. {{1,1,1,1}} {{1},{1,1,1}} {{1},{2},{1,1}} {{1},{2},{3},{4}}

{{1,1,1,2}} {{1},{1,1,2}} {{1},{2},{1,2}}

{{1,1,2,2}} {{1},{1,2,2}} {{1},{2},{1,3}}

{{1,1,2,3}} {{1},{1,2,3}} {{1},{2},{3,3}}

{{1,2,3,4}} {{1},{2,2,2}} {{1},{2},{3,4}}

{{1},{2,2,3}}

{{1},{2,3,4}}

{{1,1},{1,2}}

{{1,1},{2,2}}

{{1,1},{2,3}}

{{1,2},{1,3}}

{{1,2},{3,4}}

MATHEMATICA

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]& /@ sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];

mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s, Flatten[MapIndexed[Table[#2, {#1}]&, #]]& /@ IntegerPartitions[n]}];

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i, p[[i]]}, {i, Length[p]}])], {p, Permutations[Union@@m]}]]];

Table[Length[Union[brute /@ Select[mpm[n], UnsameQ@@#&&Length[#]==k&]]], {n, 0, 5}, {k, 0, n}]

PROG

(PARI)

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}

G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}

T(n)={[Vecrev(p) | p <- Vec(G(n))]}

{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

CROSSREFS

Row sums are A316980, connected case A319557.

For multiset partitions we have A317533, connected A322133.

Counting connected components instead of edges gives A321194.

For normal multiset partitions we have A330787, row sums A317776.

For set multipartitions we have A334550.

For set-systems we have A368096, row-sums A283877 (connected A300913).

A000110 counts set-partitions, non-isomorphic A000041.

A003465 counts covering set-systems, unlabeled A055621.

A007716 counts non-isomorphic multiset partitions, connected A007718.

A049311 counts non-isomorphic set multipartitions, connected A056156.

A058891 counts set-systems, unlabeled A000612, connected A323818.

Cf. A255903, A296122, A302545, A306005, A317532, A317775, A317794, A317795, A319560, A368094, A368095.

Sequence in context: A285072 A300454 A155112 * A256130 A257566 A345117

Adjacent sequences: A368096 A368097 A368098 * A368100 A368101 A368102

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Dec 31 2023

STATUS

approved