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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. 21
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:

  {{111}}          {{112}}          {{123}}

  {{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}}{{1}}  {{2}}{{11}}      {{1}}{{2}}{{3}}

                   {{1}}{{1}{2}}

                   {{2}}{{1}{1}}

                   {{1}}{{1}}{{2}}

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

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Last modified January 22 23:50 EST 2022. Contains 350504 sequences. (Running on oeis4.)