A318805 -id:A318805 - 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

A320796 Regular triangle where T(n,k) is the number of non-isomorphic self-dual multiset partitions of weight n with k parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 5, 7, 3, 1, 1, 7, 14, 10, 3, 1, 1, 9, 23, 24, 11, 3, 1, 1, 12, 39, 53, 34, 12, 3, 1, 1, 14, 61, 102, 86, 39, 12, 3, 1, 1, 17, 90, 193, 201, 117, 42, 12, 3, 1, 1, 20, 129, 340, 434, 310, 136, 43, 12, 3, 1, 1, 24, 184, 584, 902, 778, 412, 149, 44, 12, 3, 1

Offset: 1

Gus Wiseman, Nov 02 2018

Also the number of nonnegative integer k X k symmetric matrices with sum of elements equal to n and no zero rows or columns, up to row and column permutations.
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.

			Triangle begins:
   1
   1   1
   1   2   1
   1   4   3   1
   1   5   7   3   1
   1   7  14  10   3   1
   1   9  23  24  11   3   1
   1  12  39  53  34  12   3   1
   1  14  61 102  86  39  12   3   1
   1  17  90 193 201 117  42  12   3   1
Non-isomorphic representatives of the multiset partitions for n = 1 through 5 (commas elided):
1: {{1}}
.
2: {{11}}  {{1}{2}}
.
3: {{111}}  {{1}{22}}  {{1}{2}{3}}
.           {{2}{12}}
.
4: {{1111}}  {{11}{22}}  {{1}{1}{23}}  {{1}{2}{3}{4}}
.            {{12}{12}}  {{1}{2}{33}}
.            {{1}{222}}  {{1}{3}{23}}
.            {{2}{122}}
.
5: {{11111}}  {{11}{122}}  {{1}{22}{33}}  {{1}{2}{2}{34}}  {{1}{2}{3}{4}{5}}
.             {{11}{222}}  {{1}{23}{23}}  {{1}{2}{3}{44}}
.             {{12}{122}}  {{1}{2}{333}}  {{1}{2}{4}{34}}
.             {{1}{2222}}  {{1}{3}{233}}
.             {{2}{1222}}  {{2}{12}{33}}
.                          {{2}{13}{23}}
.                          {{3}{3}{123}}

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A316983.

Cf. A000219, A007716, A316980, A317533, A318805, A319560, A319616, A319721, A320797-A320813.

row(n)={vector(n, k, T(k,n) - T(k-1,n))} \\ T(n,k) defined in A318805. - Andrew Howroyd, Jan 16 2024

T(n,k) = A318805(k,n) - A318805(k-1,n). - Andrew Howroyd, Jan 16 2024

a(56) onwards from Andrew Howroyd, Jan 16 2024

A318795 Array read by antidiagonals: T(n,k) is the number of inequivalent nonnegative integer n X n matrices with sum of elements equal to k, under row and column permutations.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 5, 4, 1, 1, 11, 10, 4, 1, 1, 14, 24, 10, 4, 1, 1, 24, 51, 33, 10, 4, 1, 1, 30, 114, 78, 33, 10, 4, 1, 1, 45, 219, 224, 91, 33, 10, 4, 1, 1, 55, 424, 549, 277, 91, 33, 10, 4, 1, 1, 76, 768, 1403, 792, 298, 91, 33, 10, 4, 1, 1, 91, 1352, 3292, 2341, 881, 298, 91, 33, 10, 4, 1

Offset: 1

Andrew Howroyd, Sep 03 2018

			Array begins:
===========================================================
n\k| 1 2  3  4  5   6   7    8    9    10     11     12
---+-------------------------------------------------------
1  | 1 1  1  1  1   1   1    1    1     1      1      1 ...
2  | 1 4  5 11 14  24  30   45   55    76     91    119 ...
3  | 1 4 10 24 51 114 219  424  768  1352   2278   3759 ...
4  | 1 4 10 33 78 224 549 1403 3292  7677  16934  36581 ...
5  | 1 4 10 33 91 277 792 2341 6654 18802  51508 138147 ...
6  | 1 4 10 33 91 298 881 2825 8791 27947  87410 272991 ...
7  | 1 4 10 33 91 298 910 2974 9655 32287 108274 367489 ...
8  | 1 4 10 33 91 298 910 3017 9886 33767 116325 410298 ...
9  | 1 4 10 33 91 298 910 3017 9945 34124 118729 424498 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Andrew Howroyd, Additional PARI Programs

Rows 2..7 are A053307, A052365, A052366, A052367, A052372, A052373.
Main diagonal is A007716.
Cf. A214398, A246106, A304942, A318805.

permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
c[p_List, q_List, k_] := SeriesCoefficient[1/Product[(1 - x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}, {i, 1, Length[p]}], {x, 0, k}];
M[m_, n_, k_] := Module[{s=0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];
Table[M[n-k+1, n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 12 2018, after Andrew Howroyd *)

\\ see also link.
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)={1/prod(j=1, #q, (1-y^lcm(t,q[j]) + O(y*y^k))^gcd(t, q[j]))}
M(m, n, k)={my(s=0); forpart(q=m, s+=permcount(q)*polcoef(polcoef(exp(sum(t=1, n, K(q, t, k)/t*x^t) + O(x*x^n)), n), k)); s/m!}
for(n=1, 10, for(k=1, 12, print1(M(n, n, k), ", ")); print); \\ updated Andrew Howroyd, Mar 29 2020

T(n,k) = T(k,k) for n > 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.

A333737 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k up to permutations of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 5, 5, 1, 1, 1, 1, 3, 9, 12, 7, 1, 1, 1, 1, 4, 13, 33, 29, 11, 1, 1, 1, 1, 4, 20, 74, 142, 79, 15, 1, 1, 1, 1, 5, 28, 163, 556, 742, 225, 22, 1, 1, 1, 1, 5, 39, 319, 1919, 5369, 4454, 677, 30, 1, 1

Offset: 0

Andrew Howroyd, Apr 08 2020

Terms may be computed without generating each matrix by enumerating the number of matrices by column sum sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A188403. Burnside's lemma as applied in A318805 can be used to extend this method to the unlabeled case.

			Array begins:
==============================================
n\k | 0 1  2   3    4     5      6       7
----+-----------------------------------------
  0 | 1 1  1   1    1     1      1       1 ...
  1 | 1 1  1   1    1     1      1       1 ...
  2 | 1 1  2   2    3     3      4       4 ...
  3 | 1 1  3   5    9    13     20      28 ...
  4 | 1 1  5  12   33    74    163     319 ...
  5 | 1 1  7  29  142   556   1919    5793 ...
  6 | 1 1 11  79  742  5369  31781  156191 ...
  7 | 1 1 15 225 4454 64000 692599 5882230 ...
  ...
The T(3,3) = 5 matrices are:
   [0 0 3]  [0 1 2]  [0 1 2]  [1 0 2]  [1 1 1]
   [0 3 0]  [1 1 1]  [1 2 0]  [0 3 0]  [1 1 1]
   [3 0 0]  [2 1 0]  [2 0 1]  [2 0 1]  [1 1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..377
Ming Yean Lim, The number of irreducibles in the plethysm s_lambda[s_m], arXiv:2503.21108 [math.CO], 2025. See p. 8.

Rows n=0..5 are A000012, A000012, A008619, A106607, A333886, A333887.
Columns n=0..5 are A000012, A000012, A000041, A333888, A333889, A333890.
Main diagonal is A333738.
Cf. A188403 (labeled case), A333159 (binary), A333733 (not necessarily symmetric).
Cf. A318805, A333893.

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

cp's OEIS Frontend

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

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A320796 Regular triangle where T(n,k) is the number of non-isomorphic self-dual multiset partitions of weight n with k parts.

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A318795 Array read by antidiagonals: T(n,k) is the number of inequivalent nonnegative integer n X n matrices with sum of elements equal to k, under row and column permutations.

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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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A333737 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k up to permutations of rows and 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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