A320796 -id:A320796 - cp's OEIS frontend

A316983 Number of non-isomorphic self-dual multiset partitions of weight n.

1, 1, 2, 4, 9, 17, 36, 72, 155, 319, 677, 1429, 3094, 6648, 14518, 31796, 70491, 156818, 352371, 795952, 1813580, 4155367, 9594425, 22283566, 52122379, 122631874, 290432439, 691831161, 1658270316, 3997272089, 9692519896, 23631827354, 57943821449, 142834652193

Offset: 0

Gus Wiseman, Jul 18 2018

nonn

Also the number of nonnegative integer square symmetric matrices with sum of elements equal to n, under row and column permutations.

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity.

			Non-isomorphic representatives of the a(4) = 9 self-dual multiset partitions:
  (1111),
  (1)(222), (2)(122), (11)(22), (12)(12),
  (1)(1)(23), (1)(2)(33), (1)(3)(23),
  (1)(2)(3)(4).
The a(4) = 9 square symmetric matrices:
. [4]
.
. [3 0]  [2 0]  [2 1]  [1 1]
. [0 1]  [0 2]  [1 0]  [1 1]
.
. [2 0 0]  [1 1 0]  [0 1 1]
. [0 1 0]  [1 0 0]  [1 0 0]
. [0 0 1]  [0 0 1]  [1 0 0]
.
. [1 0 0 0]
. [0 1 0 0]
. [0 0 1 0]
. [0 0 0 1]

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

Row sums of A320796.
Main diagonal of A318805.
Cf. A000009, A001055, A007716, A007717, A020555, A045778.
Cf. A316974, A316978, A316979, A316980, A316981.

vector(25, n, n--; T(n,n)) \\ T(n,k) defined in A318805. - Andrew Howroyd, Jan 16 2024

Terms a(9) and beyond from Andrew Howroyd, Sep 03 2018

A138178 Number of symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

Original entry on oeis.org

1, 1, 3, 9, 33, 125, 531, 2349, 11205, 55589, 291423, 1583485, 8985813, 52661609, 319898103, 2000390153, 12898434825, 85374842121, 580479540219, 4041838056561, 28824970996809, 210092964771637, 1564766851282299, 11890096357039749, 92151199272181629

Offset: 0

Vladeta Jovovic, Mar 03 2008

Number of normal semistandard Young tableaux of size n, where a tableau is normal if its entries span an initial interval of positive integers. - Gus Wiseman, Feb 23 2018

			a(4) = 33 because there are 1 such matrix of type 1 X 1, 7 matrices of type 2 X 2, 15 of type 3 X 3 and 10 of type 4 X 4, cf. A138177.
From _Gus Wiseman_, Feb 23 2018: (Start)
The a(3) = 9 normal semistandard Young tableaux:
1   1 2   1 3   1 2   1 1   1 2 3   1 2 2   1 1 2   1 1 1
2   3     2     2     2
3
(End)
From _Gus Wiseman_, Nov 14 2018: (Start)
The a(4) = 33 matrices:
[4]
.
[30][21][20][11][10][02][01]
[01][10][02][11][03][20][12]
.
[200][200][110][101][100][100][100][100][011][010][010][010][001][001][001]
[010][001][100][010][020][011][010][001][100][110][101][100][020][010][001]
[001][010][001][100][001][010][002][011][100][001][010][002][100][101][110]
.
[1000][1000][1000][1000][0100][0100][0010][0010][0001][0001]
[0100][0100][0010][0001][1000][1000][0100][0001][0100][0010]
[0010][0001][0100][0010][0010][0001][1000][1000][0010][0100]
[0001][0010][0001][0100][0001][0010][0001][0100][1000][1000]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Row sums of A138177.
Cf. A007716, A120733, A135588, A296188.
Cf. A057151, A104601, A104602, A120732, A316983, A320796, A321401, A321405, A321407.

gf:= proc(j) local k, n; add(add((-1)^(n-k) *binomial(n, k) *(1-x)^(-k) *(1-x^2)^(-binomial(k, 2)), k=0..n), n=0..j) end: a:= n-> coeftayl(gf(n+1), x=0, n): seq(a(n), n=0..25); # Alois P. Heinz, Sep 25 2008

Table[Sum[SeriesCoefficient[1/(2^(k+1)*(1-x)^k*(1-x^2)^(k*(k-1)/2)),{x,0,n}],{k,0,Infinity}],{n,0,20}]  (* Vaclav Kotesovec, Jul 03 2014 *)
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]]; Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],Sort[Reverse/@#]==#]&]],{n,5}] (* Gus Wiseman, Nov 14 2018 *)

G.f.: Sum_{n>=0} Sum_{k=0..n} (-1)^(n-k)*C(n,k)*(1-x)^(-k)*(1-x^2)^(-C(k,2)).

G.f.: Sum_{n>=0} 2^(-n-1)*(1-x)^(-n)*(1-x^2)^(-C(n,2)). - Vladeta Jovovic, Dec 09 2009

More terms from Alois P. Heinz, Sep 25 2008

A321405 Number of non-isomorphic self-dual set systems of weight n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 6, 9, 16, 28, 47

Offset: 0

Gus Wiseman, Nov 15 2018

Also the number of (0,1) symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the rows are all different.
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(8) = 16 set systems:
  {{1}}  {{1}{2}}  {{2}{12}}    {{1}{3}{23}}    {{2}{13}{23}}
                   {{1}{2}{3}}  {{1}{2}{3}{4}}  {{1}{2}{4}{34}}
                                                {{1}{2}{3}{4}{5}}
.
  {{12}{13}{23}}        {{13}{23}{123}}          {{1}{13}{14}{234}}
  {{3}{23}{123}}        {{1}{23}{24}{34}}        {{12}{13}{24}{34}}
  {{1}{3}{24}{34}}      {{1}{4}{34}{234}}        {{1}{24}{34}{234}}
  {{2}{4}{12}{34}}      {{2}{13}{24}{34}}        {{2}{14}{34}{234}}
  {{1}{2}{3}{5}{45}}    {{3}{4}{14}{234}}        {{3}{4}{134}{234}}
  {{1}{2}{3}{4}{5}{6}}  {{1}{2}{4}{35}{45}}      {{4}{13}{14}{234}}
                        {{1}{3}{5}{23}{45}}      {{1}{2}{34}{35}{45}}
                        {{1}{2}{3}{4}{6}{56}}    {{1}{2}{5}{45}{345}}
                        {{1}{2}{3}{4}{5}{6}{7}}  {{1}{3}{24}{35}{45}}
                                                 {{1}{4}{5}{25}{345}}
                                                 {{2}{4}{12}{35}{45}}
                                                 {{4}{5}{13}{23}{45}}
                                                 {{1}{2}{3}{5}{46}{56}}
                                                 {{1}{2}{4}{6}{34}{56}}
                                                 {{1}{2}{3}{4}{5}{7}{67}}
                                                 {{1}{2}{3}{4}{5}{6}{7}{8}}

Cf. A000219, A007716, A045778, A049311, A135588, A138178, A283877, A316980, A316983, A319616.

Cf. A320796, A320797, A321401, A321403, A321406.

A055884 Euler transform of partition triangle A008284.

Original entry on oeis.org

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 *)

A320797 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 9, 15, 33, 60, 121

Offset: 0

Gus Wiseman, Nov 02 2018

Also the number of nonnegative integer square symmetric matrices with sum of elements equal to n and no rows or columns summing to 0 or 1, up to row and column permutations.

			Non-isomorphic representatives of the a(2) = 1 through a(7) = 15 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{11}{22}}  {{11}{122}}  {{111}{222}}    {{111}{1222}}
                   {{12}{12}}  {{11}{222}}  {{112}{122}}    {{111}{2222}}
                               {{12}{122}}  {{11}{2222}}    {{112}{1222}}
                                            {{12}{1222}}    {{11}{22222}}
                                            {{22}{1122}}    {{12}{12222}}
                                            {{11}{22}{33}}  {{122}{1122}}
                                            {{11}{23}{23}}  {{22}{11222}}
                                            {{12}{13}{23}}  {{11}{12}{233}}
                                                            {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{11}{23}{233}}
                                                            {{12}{12}{333}}
                                                            {{12}{13}{233}}
                                                            {{13}{23}{123}}
Inequivalent representatives of the a(6) = 9 symmetric matrices with no rows or columns summing to 1:
  [6]
.
  [3 0]  [2 1]  [4 0]  [3 1]  [2 2]
  [0 3]  [1 2]  [0 2]  [1 1]  [2 0]
.
  [2 0 0]  [2 0 0]  [1 1 0]
  [0 2 0]  [0 1 1]  [1 0 1]
  [0 0 2]  [0 1 1]  [0 1 1]

Cf. A000219, A007716, A302545, A316980, A316983, A319560, A319616, A319721.

Cf. A320796, A320798, A320799, A320804, A320811, A320812, A320813.

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.

A135588 Number of symmetric (0,1)-matrices with exactly n entries equal to 1 and no zero rows or columns.

Original entry on oeis.org

1, 1, 2, 6, 20, 74, 302, 1314, 6122, 29982, 154718, 831986, 4667070, 27118610, 163264862, 1013640242, 6488705638, 42687497378, 288492113950, 1998190669298, 14177192483742, 102856494496050, 762657487965086, 5771613810502002, 44555989658479726, 350503696871063138

Offset: 0

Vladeta Jovovic, Feb 25 2008, Mar 03 2008, Mar 04 2008

nonn

			From _Gus Wiseman_, Nov 14 2018: (Start)
The a(4) = 20 matrices:
  [11]
  [11]
.
  [110][101][100][100][011][010][010][001][001]
  [100][010][011][001][100][110][101][010][001]
  [001][100][010][011][100][001][010][101][110]
.
  [1000][1000][1000][1000][0100][0100][0010][0010][0001][0001]
  [0100][0100][0010][0001][1000][1000][0100][0001][0100][0010]
  [0010][0001][0100][0010][0010][0001][1000][1000][0010][0100]
  [0001][0010][0001][0100][0001][0010][0001][0100][1000][1000]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..721 (first 51 terms from Vincenzo Librandi)

Row sums of A135589.

Cf. A049311, A054976, A101370, A104601, A104602, A120733, A138178, A283877, A316983, A320796, A321401, A321405.

Table[Sum[SeriesCoefficient[(1+x)^k*(1+x^2)^(k*(k-1)/2)/2^(k+1),{x,0,n}],{k,0,Infinity}],{n,0,20}] (* Vaclav Kotesovec, Jul 02 2014 *)
Join[{1},  Table[Length[Select[Subsets[Tuples[Range[n], 2], {n}], And[Union[First/@#]==Range[Max@@First/@#], Union[Last/@#]==Range[Max@@Last/@#], Sort[Reverse/@#]==#]&]], {n, 5}]] (* Gus Wiseman, Nov 14 2018 *)

G.f.: Sum_{n>=0} (1+x)^n*(1+x^2)^binomial(n,2)/2^(n+1).

G.f.: Sum_{n>=0} (Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*(1+x)^k*(1+x^2)^binomial(k,2)).

A320801 Regular triangle read by rows where T(n,k) 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.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 3, 6, 0, 1, 6, 10, 16, 0, 1, 6, 20, 30, 34, 0, 1, 9, 31, 75, 92, 90, 0, 1, 9, 45, 126, 246, 272, 211, 0, 1, 12, 60, 223, 501, 839, 823, 558, 0, 1, 12, 81, 324, 953, 1900, 2762, 2482, 1430, 0, 1, 15, 100, 491, 1611, 4033, 7120, 9299, 7629, 3908

Offset: 0

Gus Wiseman, Nov 09 2018

			Triangle begins:
   1
   0   1
   0   1   3
   0   1   3   6
   0   1   6  10  16
   0   1   6  20  30  34
   0   1   9  31  75  92  90
   0   1   9  45 126 246 272 211
   0   1  12  60 223 501 839 823 558

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

Row sums are A007716. Last column is A049311.

Cf. A316980, A316983, A317533, A318805, A319616, A319721, A320796, A320808, A321194.

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)={prod(j=1, #q, my(g=gcd(t, q[j]), e=(q[j]/g)); (1 - y^e + y^e/(1 - x^e) + O(x*x^k))^g) - 1}
G(n)={my(s=0); forpart(q=n, s+=permcount(q)*exp(sum(t=1, n, substvec(K(q, t, n\t)/t, [x,y], [x^t,y^t])) + O(x*x^n))); s/n!}
T(n)=[Vecrev(p) | p<-Vec(G(n))]
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 16 2024

Offset corrected by Andrew Howroyd, Jan 16 2024

A321401 Number of non-isomorphic strict self-dual multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 29, 57, 117, 240, 498

Offset: 0

Gus Wiseman, Nov 09 2018

Also the number of nonnegative integer symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the rows (or columns) are all different.

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}}.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 14 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}        {{1,1,1,1,1}}
         {{1},{2}}  {{1},{2,2}}    {{1,1},{2,2}}      {{1,1},{1,2,2}}
                    {{2},{1,2}}    {{1},{2,2,2}}      {{1,1},{2,2,2}}
                    {{1},{2},{3}}  {{2},{1,2,2}}      {{1,2},{1,2,2}}
                                   {{1},{2},{3,3}}    {{1},{2,2,2,2}}
                                   {{1},{3},{2,3}}    {{2},{1,2,2,2}}
                                   {{1},{2},{3},{4}}  {{1},{2,2},{3,3}}
                                                      {{1},{2},{3,3,3}}
                                                      {{1},{3},{2,3,3}}
                                                      {{2},{1,2},{3,3}}
                                                      {{2},{1,3},{2,3}}
                                                      {{1},{2},{3},{4,4}}
                                                      {{1},{2},{4},{3,4}}
                                                      {{1},{2},{3},{4},{5}}

Cf. A000219, A007716, A045778, A059201, A316980, A316983, A319560, A319616.

Cf. A320796, A320797, A321402, A321405, A321406, A321407.

A321406 Number of non-isomorphic self-dual set systems of weight n with no singletons.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 1, 2, 4

Offset: 0

Gus Wiseman, Nov 15 2018

Also the number of 0-1 symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the rows are all different and none sums to 1.
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(6) = 1 through a(10) = 4 set systems:
   6: {{1,2},{1,3},{2,3}}
   7: {{1,3},{2,3},{1,2,3}}
   8: {{1,2},{1,3},{2,4},{3,4}}
   9: {{1,2},{1,3},{1,4},{2,3,4}}
   9: {{1,2},{1,4},{3,4},{2,3,4}}
  10: {{1,2},{2,4},{1,3,4},{2,3,4}}
  10: {{1,3},{2,4},{1,3,4},{2,3,4}}
  10: {{1,4},{2,4},{3,4},{1,2,3,4}}
  10: {{1,2},{1,3},{2,4},{3,5},{4,5}}

Cf. A000219, A007716, A045778, A049311, A135588, A138178, A283877, A302545, A316980, A316983.

Cf. A320796, A320797, A321401, A321402, A321403, A321405.

cp's OEIS Frontend

A316983 Number of non-isomorphic self-dual multiset partitions of weight n.

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A138178 Number of symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

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A321405 Number of non-isomorphic self-dual set systems of weight n.

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A055884 Euler transform of partition triangle A008284.

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A320797 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons.

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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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A135588 Number of symmetric (0,1)-matrices with exactly n entries equal to 1 and no zero rows or columns.

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A320801 Regular triangle read by rows where T(n,k) 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.

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A321401 Number of non-isomorphic strict self-dual multiset partitions of weight n.

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