A306186 - OEIS

A306186

Array read by antidiagonals upwards where A(n, k) is the number of non-isomorphic multiset partitions of weight n with k levels of brackets.

1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 7, 33, 21, 8, 1, 11, 91, 104, 36, 10, 1, 15, 298, 452, 238, 55, 12, 1, 22, 910, 2335, 1430, 455, 78, 14, 1, 30, 3017, 11992, 10179, 3505, 775, 105, 16, 1, 42, 9945, 66810, 74299, 31881, 7297, 1218, 136, 18, 1, 56

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..56.

EXAMPLE

Array begins:

k=1: k=2: k=3: k=4: k=5: k=6:

n=1: 1 1 1 1 1 1

n=2: 2 4 6 8 10 12

n=3: 3 10 21 36 55 78

n=4: 5 33 104 238 455 775

n=5: 7 91 452 1430 3505 7297

n=6: 11 298 2335 10179 31881 80897

Non-isomorphic representatives of the A(3,3) = 21 multiset partitions:

{{1}{11}} {{1}{12}} {{1}{23}}

{{1}}{{11}} {{2}{11}} {{1}}{{23}}

{{1}{1}{1}} {{1}}{{12}} {{1}{2}{3}}

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

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

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

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]]]];

undats[m_]:=Union[DeleteCases[Cases[m, _?AtomQ, {0, Infinity}, Heads->True], List]];

expnorm[m_]:=If[Length[undats[m]]==0, m, If[undats[m]!=Range[Max@@undats[m]], expnorm[m/.Apply[Rule, Table[{undats[m][[i]], i}, {i, Length[undats[m]]}], {1}]], First[Sort[expnorm[m, 1]]]]];

expnorm[m_, aft_]:=If[Length[undats[m]]<=aft, {m}, With[{mx=Table[Count[m, i, {0, Infinity}, Heads->True], {i, Select[undats[m], #1>=aft&]}]}, Union@@(expnorm[#1, aft+1]&)/@Union[Table[MapAt[Sort, m/.{par+aft-1->aft, aft->par+aft-1}, Position[m, _[___]]], {par, First/@Position[mx, Max[mx]]}]]]];

strnorm[n_]:=(Flatten[MapIndexed[Table[#2, {#1}]&, #1]]&)/@IntegerPartitions[n];

kmp[n_, k_]:=kmp[n, k]=If[k==1, strnorm[n], Union[expnorm/@Join@@mps/@kmp[n, k-1]]];

Table[Length[kmp[sum-k, k]], {sum, 1, 7}, {k, 1, sum-1}]

CROSSREFS

Columns: A000041 (k=1), A007716 (k=2), A318566 (k=3).

Rows: A000012 (n=1), A005843 (n=2), A014105 (n=3).

Cf. A002846, A096751, A144150, A290353, A317533, A317791, A323718, A323719.

Sequence in context: A055888 A094442 A060642 * A154929 A249042 A262472

Adjacent sequences: A306183 A306184 A306185 * A306187 A306188 A306189

KEYWORD

nonn,tabl,more

AUTHOR

Gus Wiseman, Jan 27 2019

EXTENSIONS

a(46)-a(56) from Robert Price, May 11 2021

STATUS

approved