A320801 -id:A320801 - cp's OEIS frontend

A055884 Euler transform of partition triangle A008284.

1, 1, 2, 1, 2, 3, 1, 4, 4, 5, 1, 4, 8, 7, 7, 1, 6, 12, 16, 12, 11, 1, 6, 17, 25, 28, 19, 15, 1, 8, 22, 43, 49, 48, 30, 22, 1, 8, 30, 58, 87, 88, 77, 45, 30, 1, 10, 36, 87, 134, 167, 151, 122, 67, 42, 1, 10, 45, 113, 207, 270, 296, 247, 185, 97, 56, 1, 12, 54, 155, 295, 448, 510, 507, 394, 278, 139, 77

Offset: 1

Christian G. Bower, Jun 09 2000

Number of multiset partitions of length-k integer partitions of n. - Gus Wiseman, Nov 09 2018

			From _Gus Wiseman_, Nov 09 2018: (Start)
Triangle begins:
   1
   1   2
   1   2   3
   1   4   4   5
   1   4   8   7   7
   1   6  12  16  12  11
   1   6  17  25  28  19  15
   1   8  22  43  49  48  30  22
   1   8  30  58  87  88  77  45  30
   ...
The fifth row {1, 4, 8, 7, 7} counts the following multiset partitions:
  {{5}}   {{1,4}}     {{1,1,3}}       {{1,1,1,2}}         {{1,1,1,1,1}}
          {{2,3}}     {{1,2,2}}      {{1},{1,1,2}}       {{1},{1,1,1,1}}
         {{1},{4}}   {{1},{1,3}}     {{1,1},{1,2}}       {{1,1},{1,1,1}}
         {{2},{3}}   {{1},{2,2}}     {{2},{1,1,1}}      {{1},{1},{1,1,1}}
                     {{2},{1,2}}    {{1},{1},{1,2}}     {{1},{1,1},{1,1}}
                     {{3},{1,1}}    {{1},{2},{1,1}}    {{1},{1},{1},{1,1}}
                    {{1},{1},{3}}  {{1},{1},{1},{2}}  {{1},{1},{1},{1},{1}}
                    {{1},{2},{2}}
(End)

Alois P. Heinz, Rows n = 1..200, flattened
N. J. A. Sloane, Transforms

Row sums give A001970.
Main diagonal gives A000041.
Columns k=1-2 give: A057427, A052928.
T(n+2,n+1) gives A000070.
T(2n,n) gives A360468.
Cf. A055885, A055886, A360742.
Cf. A000219, A007716, A008284, A255906, A317449, A317532, A317533, A320796, A320801, A320808.

h:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i)))))
    end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i)+k-1, k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n,k), k=1..n), n=1..12);  # Alois P. Heinz, Feb 17 2023

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]]]];
Table[Length[Join@@mps/@IntegerPartitions[n,{k}]],{n,5},{k,n}] (* Gus Wiseman, Nov 09 2018 *)

A321449 Regular triangle read by rows where T(n,k) is the number of twice-partitions of n with a combined total of k parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 4, 5, 5, 0, 1, 4, 8, 8, 7, 0, 1, 6, 13, 19, 16, 11, 0, 1, 6, 17, 27, 32, 24, 15, 0, 1, 8, 24, 47, 61, 62, 41, 22, 0, 1, 8, 30, 63, 99, 111, 100, 61, 30, 0, 1, 10, 38, 94, 158, 209, 210, 170, 95, 42, 0, 1, 10, 45, 119, 229, 328, 382, 348, 259, 136, 56

Offset: 0

Gus Wiseman, Nov 10 2018

A twice partition of n (A063834) is a choice of an integer partition of each part in an integer partition of n.

			Triangle begins:
   1
   0   1
   0   1   2
   0   1   2   3
   0   1   4   5   5
   0   1   4   8   8   7
   0   1   6  13  19  16  11
   0   1   6  17  27  32  24  15
   0   1   8  24  47  61  62  41  22
   0   1   8  30  63  99 111 100  61  30
The sixth row {0, 1, 6, 13, 19, 16, 11} counts the following twice-partitions:
  (6)  (33)    (222)      (2211)        (21111)          (111111)
       (42)    (321)      (3111)        (1111)(2)        (111)(111)
       (51)    (411)      (111)(3)      (111)(21)        (1111)(11)
       (3)(3)  (21)(3)    (211)(2)      (21)(111)        (11111)(1)
       (4)(2)  (22)(2)    (21)(21)      (211)(11)        (11)(11)(11)
       (5)(1)  (31)(2)    (22)(11)      (2111)(1)        (111)(11)(1)
               (3)(21)    (221)(1)      (11)(11)(2)      (1111)(1)(1)
               (32)(1)    (3)(111)      (111)(2)(1)      (11)(11)(1)(1)
               (4)(11)    (31)(11)      (11)(2)(11)      (111)(1)(1)(1)
               (41)(1)    (311)(1)      (2)(11)(11)      (11)(1)(1)(1)(1)
               (2)(2)(2)  (11)(2)(2)    (21)(11)(1)      (1)(1)(1)(1)(1)(1)
               (3)(2)(1)  (2)(11)(2)    (211)(1)(1)
               (4)(1)(1)  (21)(2)(1)    (11)(2)(1)(1)
                          (2)(2)(11)    (2)(11)(1)(1)
                          (22)(1)(1)    (21)(1)(1)(1)
                          (3)(11)(1)    (2)(1)(1)(1)(1)
                          (31)(1)(1)
                          (2)(2)(1)(1)
                          (3)(1)(1)(1)

Alois P. Heinz, Rows n = 0..200, flattened

Row sums are A063834. Last column is A000041.

Cf. A000219, A001970, A007716, A008284, A055884, A289501, A317449, A317532, A317533, A320801, A320808.

g:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      g(n, i-1)+ `if`(i>n, 0, expand(g(n-i, i)*x)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, i-1)+ `if`(i>n, 0, expand(b(n-i, i)*g(i$2))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Nov 11 2018

Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@ptn],Length[Join@@#]==k&],{ptn,IntegerPartitions[n]}]],{n,0,10},{k,0,n}]
(* Second program: *)
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, x^n,
     g[n, i - 1] + If[i > n, 0, Expand[g[n - i, i]*x]]];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n,
     b[n, i - 1] + If[i > n, 0, Expand[b[n - i, i]*g[i, i]]]];
T[n_] := CoefficientList[b[n, n], x];
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

O.g.f.: Product_{n >= 0} 1/(1 - x^n * (Sum_{0 <= k <= n} A008284(n,k) * t^k)).

A320808 Regular tetrangle where T(n,k,i) is the number of nonnegative integer matrices up to row and column permutations with no zero rows or columns and k nonzero entries summing to n, with i columns.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0, 1, 0, 2, 4, 0, 1, 5, 4, 0, 1, 5, 5, 5, 0, 0, 1, 0, 2, 4, 0, 2, 10, 8, 0, 1, 9, 13, 7, 0, 1, 5, 12, 9, 7, 0, 0, 1, 0, 3, 6, 0, 3, 16, 12, 0, 2, 24, 33, 16, 0, 1, 14, 36, 29, 12, 0, 1, 9, 23, 29

Offset: 1

Gus Wiseman, Nov 09 2018

			Tetrangle begins:
  1  0    0      0        0          0
     0 1  0 1    0 1      0 1        0 1
          0 1 2  0 1 2    0 2 4      0 2 4
                 0 1 2 3  0 1 5 4    0 2 10 8
                          0 1 5 5 5  0 1 9 13 7
                                     0 1 5 12 9 7

Triangle sums are A007716. Triangle of row sums is A320801. Triangle of column sums is A317533. Triangle of last columns (without its leading column 1,0,0,0,...) is A055884.

Cf. A000219, A008284, A316980, A316983, A318805, A319616, A320796, A321194.

A321760 Number of non-isomorphic multiset partitions of weight n with no constant parts or vertices that appear in only one part.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 7, 9, 37, 79, 273, 755, 2648, 8432, 29872, 104624, 384759, 1432655, 5502563, 21533141, 86291313, 352654980, 1471073073, 6253397866, 27083003687, 119399628021, 535591458635, 2443030798539, 11326169401988, 53343974825122, 255121588496338

Offset: 0

Gus Wiseman, Nov 29 2018

nonn

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n in which every row and column has at least two nonzero entries.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(4) = 1 through a(7) = 9 multiset partitions:
  {{1,2},{1,2}}  {{1,2},{1,2,2}}  {{1,1,2},{1,2,2}}    {{1,1,2},{1,2,2,2}}
                                  {{1,2},{1,1,2,2}}    {{1,2},{1,1,2,2,2}}
                                  {{1,2},{1,2,2,2}}    {{1,2},{1,2,2,2,2}}
                                  {{1,2,2},{1,2,2}}    {{1,2,2},{1,1,2,2}}
                                  {{1,2,3},{1,2,3}}    {{1,2,2},{1,2,2,2}}
                                  {{1,2},{1,2},{1,2}}  {{1,2,3},{1,2,3,3}}
                                  {{1,2},{1,3},{2,3}}  {{1,2},{1,2},{1,2,2}}
                                                       {{1,2},{1,3},{2,3,3}}
                                                       {{1,3},{2,3},{1,2,3}}

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

Row sums of A369286.

Cf. A001970, A007716, A050535, A055884, A120733, A317533, A320665, A320798, A320801, A320808, A321407.

Vec(G(20,1)) \\ G defined in A369286. - Andrew Howroyd, Jan 28 2024

a(11) onwards from Andrew Howroyd, Jan 27 2024

A320800 Number of non-isomorphic multiset partitions of weight n in which both the multiset union of the parts and the multiset union of the dual parts are aperiodic.

Original entry on oeis.org

1, 1, 1, 5, 14, 78, 157, 881, 2267, 9257, 28397

Offset: 0

Gus Wiseman, Nov 02 2018

The latter condition is equivalent to the parts having relatively prime sizes.
A multiset is aperiodic if its multiplicities are relatively prime.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 14 multiset partitions:
  {{1}}  {{1},{2}}  {{1},{2,2}}    {{1},{2,2,2}}
                    {{1},{2,3}}    {{1},{2,3,3}}
                    {{2},{1,2}}    {{1},{2,3,4}}
                    {{1},{2},{2}}  {{2},{1,2,2}}
                    {{1},{2},{3}}  {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303431, A303546, A303547, A316983, A320801-A320810.

A321407 Number of non-isomorphic multiset partitions of weight n with no constant parts.

Original entry on oeis.org

1, 0, 1, 2, 7, 13, 47, 111, 367, 1057, 3474, 11116, 38106, 131235, 470882, 1720959, 6472129, 24860957, 97779665, 392642763, 1610045000, 6732768139, 28699327441, 124600601174, 550684155992, 2476019025827, 11320106871951, 52598300581495, 248265707440448, 1189855827112636, 5787965846277749

Offset: 0

Gus Wiseman, Nov 29 2018

nonn

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which every row has at least two nonzero entries.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 multiset partitions:
  {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}
                      {{1,2},{1,2}}  {{1,2,3,4,4}}
                      {{1,2},{3,4}}  {{1,2,3,4,5}}
                      {{1,3},{2,3}}  {{1,2},{1,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,3},{1,2,3}}

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

Cf. A001970, A007716, A050535, A055884, A120733, A317533, A320798, A320801, A320808, A321760.

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))}
S(q, t, k)={sum(j=1, #q, if(t%q[j]==0, q[j]))*vector(k,i,1)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(sum(t=1, n, subst(x*Ser(K(q, t, n\t)-S(q, t, n\t))/t, x, x^t) )), n)); s/n!)} \\ Andrew Howroyd, Jan 17 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023

cp's OEIS Frontend

A055884 Euler transform of partition triangle A008284.

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A321449 Regular triangle read by rows where T(n,k) is the number of twice-partitions of n with a combined total of k parts.

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A320808 Regular tetrangle where T(n,k,i) is the number of nonnegative integer matrices up to row and column permutations with no zero rows or columns and k nonzero entries summing to n, with i columns.

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A321760 Number of non-isomorphic multiset partitions of weight n with no constant parts or vertices that appear in only one part.

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A320800 Number of non-isomorphic multiset partitions of weight n in which both the multiset union of the parts and the multiset union of the dual parts are aperiodic.

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A321407 Number of non-isomorphic multiset partitions of weight n with no constant parts.

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