A323719 -id:A323719 - cp's OEIS frontend

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

Offset: 1

Gus Wiseman, Jan 27 2019

			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}}

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.

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}]

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

A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 7, 1, 1, 1, 6, 15, 26, 24, 11, 1, 1, 1, 7, 21, 45, 59, 48, 15, 1, 1, 1, 8, 28, 71, 120, 140, 86, 22, 1, 1, 1, 9, 36, 105, 216, 326, 307, 160, 30, 1, 1, 1, 10, 45, 148, 357, 657, 835, 684, 282, 42, 1

Offset: 0

Paul D. Hanna, Jul 07 2004

Main diagonal forms A096752. Antidiagonal sums form A096753. Row with index n lists the row sums of the n-th matrix power of triangle A096651, for n>=0.

			n-th row lists n-dimensional partitions; table begins with n=0:
  [1,1,1,1,1,1,1,1,1,1,1,1,...],
  [1,1,2,3,5,7,11,15,22,30,42,56,...],
  [1,1,3,6,13,24,48,86,160,282,500,859,...],
  [1,1,4,10,26,59,140,307,684,1464,3122,...],
  [1,1,5,15,45,120,326,835,2145,5345,...],
  [1,1,6,21,71,216,657,1907,5507,15522,...],
  [1,1,7,28,105,357,1197,3857,12300,38430,...],
  [1,1,8,36,148,554,2024,7134,24796,84625,...],
  [1,1,9,45,201,819,3231,12321,46209,170370,...],
  [1,1,10,55,265,1165,4927,20155,80920,...],...
Array begins:
      k=0:  k=1:  k=2:  k=3:  k=4:  k=5:  k=6:  k=7:  k=8:
  n=0:  1     1     1     1     1     1     1     1     1
  n=1:  1     1     2     3     5     7    11    15    22
  n=2:  1     1     3     6    13    24    48    86   160
  n=3:  1     1     4    10    26    59   140   307   684
  n=4:  1     1     5    15    45   120   326   835  2145
  n=5:  1     1     6    21    71   216   657  1907  5507
  n=6:  1     1     7    28   105   357  1197  3857 12300
  n=7:  1     1     8    36   148   554  2024  7134 24796
  n=8:  1     1     9    45   201   819  3231 12321 46209
  n=9:  1     1    10    55   265  1165  4927 20155 80920

G. E. Andrews, The Theory of Partitions, Add.-Wes. 1976, pp. 189-197.

Pontus von Brömssen, Table of n, a(n) for n = 0..275 (first 23 antidiagonals)
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]

Rows: A000012 (n=0), A000041 (n=1), A000219 (n=2), A000293 (n=3), A000334 (n=4), A000390 (n=5), A000416 (n=6), A000427 (n=7), A179855 (n=8).
Columns: A008778 (k=4), A008779 (k=5), A042984 (k=6).
Cf. A096651, A096752, A096753.
Cf. A096806.
Cf. A042984.
Cf. A144150, A213427, A290353, A323718, A323719.

trans[x_]:=If[x=={},{},Transpose[x]];
levptns[n_,k_]:=If[k==1,IntegerPartitions[n],Join@@Table[Select[Tuples[levptns[#,k-1]&/@y],And@@(GreaterEqual@@@trans[Flatten/@(PadRight[#,ConstantArray[n,k-1]]&/@#)])&],{y,IntegerPartitions[n]}]];
Table[If[sum==k,1,Length[levptns[k,sum-k]]],{sum,0,10},{k,0,sum}] (* Gus Wiseman, Jan 27 2019 *)

T(0, n)=T(n, 0)=T(n, 1)=1 for n>=0.

Inverse binomial transforms of the columns is given by triangle A096806.

A323718 Array read by antidiagonals upwards where A(n,k) is the number of k-times partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 5, 6, 4, 1, 1, 1, 7, 15, 10, 5, 1, 1, 1, 11, 28, 34, 15, 6, 1, 1, 1, 15, 66, 80, 65, 21, 7, 1, 1, 1, 22, 122, 254, 185, 111, 28, 8, 1, 1, 1, 30, 266, 604, 739, 371, 175, 36, 9, 1, 1, 1, 42, 503, 1785, 2163, 1785, 672, 260, 45, 10, 1, 1

Offset: 0

Gus Wiseman, Jan 25 2019

A k-times partition of n for k > 1 is a sequence of (k-1)-times partitions, one of each part in an integer partition of n. A 1-times partition of n is just an integer partition of n, and the only 0-times partition of n is the number n itself.

			Array begins:
       k=0:   k=1:   k=2:   k=3:   k=4:   k=5:
  n=0:  1      1      1      1      1      1
  n=1:  1      1      1      1      1      1
  n=2:  1      2      3      4      5      6
  n=3:  1      3      6     10     15     21
  n=4:  1      5     15     34     65    111
  n=5:  1      7     28     80    185    371
  n=6:  1     11     66    254    739   1785
  n=7:  1     15    122    604   2163   6223
  n=8:  1     22    266   1785   8120  28413
  n=9:  1     30    503   4370  24446 101534
The A(4,2) = 15 twice-partitions:
  (4)  (31)    (22)    (211)      (1111)
       (3)(1)  (2)(2)  (11)(2)    (11)(11)
                       (2)(11)    (111)(1)
                       (21)(1)    (11)(1)(1)
                       (2)(1)(1)  (1)(1)(1)(1)

Alois P. Heinz, Rows n = 0..140, flattened

Columns: A000012 (k=0), A000041 (k=1), A063834 (k=2), A301595 (k=3).
Rows: A000027 (n=2), A000217 (n=3), A006003 (n=4).
Main diagonal gives A306187.
Cf. A001970, A055884, A096751, A144150, A196545, A281113, A289501, A290353, A300383, A323719, A327618, A327639.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
      1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(d-k, k), k=0..d), d=0..14);  # Alois P. Heinz, Jan 25 2019

ptnlev[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Tuples[ptnlev[#,k-1]&/@ptn],{ptn,IntegerPartitions[n]}]];
Table[Length[ptnlev[sum-k,k]],{sum,0,12},{k,0,sum}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1, 1,
     b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]];
A[n_, k_] := b[n, n, k];
Table[Table[A[d - k, k], {k, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

Column k is the formal power product transform of column k-1, where the formal power product transform of a sequence q with offset 1 is the sequence whose ordinary generating function is Product_{n >= 1} 1/(1 - q(n) * x^n).

A(n,k) = Sum_{i=0..k} binomial(k,i) * A327639(n,i). - Alois P. Heinz, Sep 20 2019

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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.

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A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

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A323718 Array read by antidiagonals upwards where A(n,k) is the number of k-times partitions of n.

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