A331485 -id:A331485 - cp's OEIS frontend

A007717 Number of symmetric polynomial functions of degree n of a symmetric matrix (of indefinitely large size) under joint row and column permutations. Also number of multigraphs with n edges (allowing loops) on an infinite set of nodes.

1, 2, 7, 23, 79, 274, 1003, 3763, 14723, 59663, 250738, 1090608, 4905430, 22777420, 109040012, 537401702, 2723210617, 14170838544, 75639280146, 413692111521, 2316122210804, 13261980807830, 77598959094772, 463626704130058, 2826406013488180, 17569700716557737

Offset: 0

Colin Mallows

nonn

Euler transform of A007719.
Also the number of non-isomorphic multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018
Number of distinct n X 2n matrices with integer entries and rows sums 2, up to row and column permutations. - Andrew Howroyd, Sep 06 2018
a(n) is the number of unlabeled loopless multigraphs with n edges rooted at one vertex. - Andrew Howroyd, Nov 22 2020

			a(2) = 7 (here - denotes an edge, = denotes a pair of parallel edges and o is a loop):
  oo
  o o
  o-
  o -
  =
  --
  - -
From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(2) = 7 multiset partitions of {1, 1, 2, 2}:
  (1122),
  (1)(122), (11)(22), (12)(12),
  (1)(1)(22), (1)(2)(12),
  (1)(1)(2)(2).
(End)
From _Gus Wiseman_, Jan 08 2024: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(3) = 7 rooted loopless multigraphs (root shown as singleton):
  {{1}}  {{1},{1,2}}  {{1},{1,2},{1,2}}
         {{1},{2,3}}  {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{1,2},{3,4}}
                      {{1},{2,3},{2,3}}
                      {{1},{2,3},{2,4}}
                      {{1},{2,3},{4,5}}
(End)

Huaien Li and David C. Torney, Enumerations of Multigraphs, 2002.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Mateo Díaz, Dmitriy Drusvyatskiy, Jack Kendrick, and Rekha R. Thomas, Invariant Kernels: Rank Stabilization and Generalization Across Dimensions, arXiv:2502.01886 [math.OC], 2025. See p. 43.
Huaien Li and David C. Torney, Enumeration of unlabelled multigraphs, Ars Combin. 75 (2005) 171-188. MR2133219.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) table 67.

Row n=2 of A331485.

Cf. A000664, A002620, A007716, A007719, A020555, A050531, A050532, A050535, A052171, A053418, A053419, A094574, A316972, A316974, A318951, A339065.

permcount[v_] := 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];
Kq[q_, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
RowSumMats[n_, m_, k_] := Module[{s=0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[Kq[q, t, k]/t x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
a[n_] := RowSumMats[n, 2n, 2];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 25}] (* Jean-François Alcover, Oct 27 2018, after Andrew Howroyd *)

\\ See A318951 for RowSumMats
a(n)=RowSumMats(n, 2*n, 2); \\ Andrew Howroyd, Sep 06 2018

\\ See A339065 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1]))} \\ Andrew Howroyd, Nov 22 2020

More terms from Vladeta Jovovic, Jan 26 2000

a(0)=1 prepended and a(16)-a(25) added by Max Alekseyev, Jun 21 2011

A306017 Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.

Original entry on oeis.org

1, 1, 4, 6, 17, 14, 66, 30, 189, 222, 550, 112, 4696, 202, 5612, 30914, 63219, 594, 453125, 980, 3602695, 5914580, 1169348, 2510, 299083307, 232988061, 23248212, 2669116433, 14829762423, 9130, 170677509317, 13684, 1724710753084, 2199418340875, 14184712185, 38316098104262

Offset: 0

Gus Wiseman, Jun 17 2018

nonn

A multiset partition of weight n is a finite multiset of finite nonempty multisets whose sizes sum to n.
Number of distinct nonnegative integer matrices with all row sums equal and total sum n up to row and column permutations. - Andrew Howroyd, Sep 05 2018
From Gus Wiseman, Oct 11 2018: (Start)
Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears the same number of times. For n = 4, non-isomorphic representatives of these 17 multiset partitions are:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,3,4}}
  {{1},{1,1,1}}
  {{1},{1,2,2}}
  {{1},{2,3,4}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1,2},{3,4}}
  {{1},{1},{1,1}}
  {{1},{1},{2,2}}
  {{1},{2},{1,2}}
  {{1},{2},{3,4}}
  {{1},{1},{1},{1}}
  {{1},{1},{2},{2}}
  {{1},{2},{3},{4}}
(End)

			Non-isomorphic representatives of the a(4) = 17 multiset partitions:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,2,2}}
  {{1,2,3,3}}
  {{1,2,3,4}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1,2},{2,2}}
  {{1,2},{3,3}}
  {{1,2},{3,4}}
  {{1,3},{2,3}}
  {{1},{1},{1},{1}}
  {{1},{1},{2},{2}}
  {{1},{2},{2},{2}}
  {{1},{2},{3},{3}}
  {{1},{2},{3},{4}}

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

Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306018, A306019, A306020, A306021, A318951.

Cf. A064573, A279787, A305551, A319616, A319056, A331485.

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];
K[q_List, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[K[q, t, k]/t*x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
a[n_] := a[n] = If[n==0, 1, If[PrimeQ[n], 2 PartitionsP[n], Sum[ RowSumMats[ n/d, n, d], {d, Divisors[n]}]]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 07 2019, after Andrew Howroyd *)

\\ See A318951 for RowSumMats.
a(n)={sumdiv(n,d,RowSumMats(n/d,n,d))} \\ Andrew Howroyd, Sep 05 2018

For p prime, a(p) = 2*A000041(p).

a(n) = Sum_{d|n} A331485(n/d, d). - Andrew Howroyd, Feb 09 2020

Terms a(11) and beyond from Andrew Howroyd, Sep 05 2018

A331315 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k columns and any number of nonzero rows with column sums n and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 14, 4, 1, 1, 8, 150, 128, 8, 1, 1, 16, 2210, 10848, 1288, 16, 1, 1, 32, 41642, 1796408, 933448, 13472, 32, 1, 1, 64, 956878, 491544512, 1852183128, 85862144, 143840, 64, 1, 1, 128, 25955630, 200901557728, 7805700498776, 2098614254048, 8206774496, 1556480, 128, 1

Offset: 0

Andrew Howroyd, Jan 13 2020

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.

A(n,k) is the number of n-uniform k-block multisets of multisets.

			Array begins:
====================================================================
n\k | 0  1      2          3                4                  5
----+---------------------------------------------------------------
  0 | 1  1      1          1                1                  1 ...
  1 | 1  1      2          4                8                 16 ...
  2 | 1  2     14        150             2210              41642 ...
  3 | 1  4    128      10848          1796408          491544512 ...
  4 | 1  8   1288     933448       1852183128      7805700498776 ...
  5 | 1 16  13472   85862144    2098614254048 140102945876710912 ...
  6 | 1 32 143840 8206774496 2516804131997152 ...
     ...
The A(2,2) = 14 matrices are:
  [1 0]  [1 0]  [1 0]  [2 0]  [1 1]  [1 0]  [1 0]
  [1 0]  [0 1]  [0 1]  [0 1]  [1 0]  [1 1]  [1 0]
  [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [0 1]  [0 2]
  [0 1]  [0 1]  [1 0]
.
  [1 0]  [1 0]  [2 1]  [2 0]  [1 1]  [1 0]  [2 2]
  [0 2]  [0 1]  [0 1]  [0 2]  [1 1]  [1 2]
  [1 0]  [1 1]

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

Rows n=1..2 are A000012, A121227.
Columns k=0..2 are A000012, A011782, A331397.
The version with binary entries is A330942.
The version with distinct columns is A331278.
Other variations considering distinct rows and columns and equivalence under different combinations of permutations of rows and columns are:
All solutions: A316674 (all), A331568 (distinct rows).
Up to row permutation: A219727, A219585, A331161, A331160.
Up to column permutation: this sequence, A331572, A331278, A331570.
Nonisomorphic: A331485.
Cf. A317583.

T(n, k)={my(m=n*k); sum(j=0, m, binomial(binomial(j+n-1, n)+k-1, k)*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))}

A(n,k) = Sum_{j=0..n*k} binomial(binomial(j+n-1,n)+k-1, k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A316674(n, j)/k!.
A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331278(n, j).
A(n, k) = A011782(n) * A330942(n, k) for k > 0.
A317583(n) = Sum_{d|n} A(n/d, d).

A331461 Array read by antidiagonals: A(n,k) is the number of nonequivalent binary matrices with k columns and any number of nonzero rows with n ones in every column up to permutation of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 5, 8, 4, 1, 1, 1, 7, 23, 16, 5, 1, 1, 1, 11, 66, 93, 30, 6, 1, 1, 1, 15, 212, 652, 332, 50, 7, 1, 1, 1, 22, 686, 6369, 6414, 1062, 80, 8, 1, 1, 1, 30, 2389, 79568, 226041, 56712, 3117, 120, 9, 1, 1, 1, 42, 8682, 1256425, 12848128, 7295812, 441881, 8399, 175, 10, 1, 1

Offset: 0

Andrew Howroyd, Jan 18 2020

A(n,k) is the number of non-isomorphic set multipartitions (multiset of sets) with k parts each part has size n.

			Array begins:
===========================================================
n\k | 0 1 2   3    4       5          6              7
----+-----------------------------------------------------
  0 | 1 1 1   1    1       1          1              1 ...
  1 | 1 1 2   3    5       7         11             15 ...
  2 | 1 1 3   8   23      66        212            686 ...
  3 | 1 1 4  16   93     652       6369          79568 ...
  4 | 1 1 5  30  332    6414     226041       12848128 ...
  5 | 1 1 6  50 1062   56712    7295812     1817321457 ...
  6 | 1 1 7  80 3117  441881  195486906   200065951078 ...
  7 | 1 1 8 120 8399 3006771 4298181107 17131523059493 ...
  ...
The A(2,3) = 8 matrices are:
  [1 0 0]  [1 1 0]  [1 1 1]  [1 1 0]  [1 1 0]  [1 1 1]  [1 1 0]  [1 1 1]
  [1 0 0]  [1 0 0]  [1 0 0]  [1 1 0]  [1 0 1]  [1 1 0]  [1 0 1]  [1 1 1]
  [0 1 0]  [0 1 0]  [0 1 0]  [0 0 1]  [0 1 0]  [0 0 1]  [0 1 1]
  [0 1 0]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]
  [0 0 1]  [0 0 1]
  [0 0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..152

Rows n=0..6 are A000012, A000041, A050535, A050913, A058783, A058784, A058785.
Columns k=0..4 are A000012, A000012, A000027(n+1), A002624, A331720.
Cf. A188392, A262809, A304942, A306018, A330942, A331485, A331508, A331509, A331510.

\\ See A304942 for Blocks
T(n,k)={Blocks(k, n*k, n)}
{ for(n=0, 7, for(k=0, 6, print1(T(n,k), ", ")); print) }

A306018(n) = Sum_{d|n} A(n/d, d).

A316674 Number A(n,k) of paths from {0}^k to {n}^k that always move closer to {n}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 13, 26, 4, 1, 1, 75, 818, 252, 8, 1, 1, 541, 47834, 64324, 2568, 16, 1, 1, 4683, 4488722, 42725052, 5592968, 26928, 32, 1, 1, 47293, 617364026, 58555826884, 44418808968, 515092048, 287648, 64, 1

Offset: 0

Alois P. Heinz, Jul 10 2018

A(n,k) is the number of nonnegative integer matrices with k columns and any number of nonzero rows with column sums n. - Andrew Howroyd, Jan 23 2020

			Square array A(n,k) begins:
  1,  1,     1,         1,              1,                    1, ...
  1,  1,     3,        13,             75,                  541, ...
  1,  2,    26,       818,          47834,              4488722, ...
  1,  4,   252,     64324,       42725052,          58555826884, ...
  1,  8,  2568,   5592968,    44418808968,      936239675880968, ...
  1, 16, 26928, 515092048, 50363651248560, 16811849850663255376, ...

Alois P. Heinz, Antidiagonals n = 0..48, flattened

Columns k=0..3 give: A000012, A011782, A052141, A316673.
Rows n=0..2 give: A000012, A000670, A059516.
Main diagonal gives A316677.
Cf. A011782, A219727, A262809, A331315, A331485, A331636.

A:= (n, k)-> `if`(k=0, 1, ceil(2^(n-1))*add(add((-1)^i*
     binomial(j, i)*binomial(j-i, n)^k, i=0..j), j=0..k*n)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

A[n_, k_] := Sum[If[k == 0, 1, Binomial[j+n-1, n]^k] Sum[(-1)^(i-j)* Binomial[i, j], {i, j, n k}], {j, 0, n k}];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Nov 04 2021 *)

T(n,k)={my(m=n*k); sum(j=0, m, binomial(j+n-1,n)^k*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))} \\ Andrew Howroyd, Jan 23 2020

A(n,k) = A262809(n,k) * A011782(n) for k>0, A(n,0) = 1.

A(n,k) = Sum_{j=0..n*k} binomial(j+n-1,n)^k * Sum_{i=j..n*k} (-1)^(i-j) * binomial(i,j). - Andrew Howroyd, Jan 23 2020

A058194 Number of n-rowed matrices with entries {0,1,2,3} and all row sums 3, up to row and column permutation.

Original entry on oeis.org

1, 3, 21, 162, 1636, 19977, 298416, 5300296, 110219750, 2639842989, 71859429837, 2198244062193, 74860247277672, 2815351714711122, 116130005180284423, 5222901881792429337, 254791333526874348652, 13420798291405599027605, 760201936044714899316798, 46137860613934391781325337, 2990483661567310913388458734

Offset: 0

Vladeta Jovovic, Nov 27 2000

nonn

Row n=3 of A331485.

Cf. A007717, A058407, A058409, A050913.

\\ See A318951 for RowSumMats
a(n)=RowSumMats(n, 3*n, 3); \\ Andrew Howroyd, Sep 06 2018

a(11)-a(20) from Andrew Howroyd, Sep 06 2018

A331721 Number of nonequivalent n-column nonnegative integer matrices with column sums 4 and any number of nonzero rows up to permutation of rows and columns.

Original entry on oeis.org

1, 5, 66, 1338, 43686, 2142277, 149056260, 13991691663, 1705750869937, 262268740077763, 49694977078927305, 11388565848749818055, 3107918736854019004136, 996840601151518793573156, 371595543159269706775806375, 159437222426663041455988930042, 78072200940916732097886063687252

Offset: 0

Andrew Howroyd, Jan 31 2020

nonn

Row n=4 of A331485.

A331722 Number of nonequivalent 2-column nonnegative integer matrices with column sums n and any number of nonzero rows up to permutation of rows and columns.

Original entry on oeis.org

1, 2, 7, 21, 66, 192, 565, 1579, 4348, 11582, 30205, 76736, 191120, 465920, 1115572, 2623885, 6074353, 13849021, 31137381, 69082779, 151390617, 327897173, 702426987, 1489108503, 3125821257, 6500086120, 13396645887, 27376083598, 55490121271, 111604327066, 222798559100

Offset: 0

Andrew Howroyd, Jan 31 2020

nonn

Zarif Ahsan, Xiran Liu, and Noah Rosenberg, Dissimilarity between a pair of draws from discrete probability vectors on finite sets of objects, Stanford Univ., 2023. See also arXiv:2410.00221, [math.CO], 2024, p. 5.

Column k=2 of A331485.

A331723 Number of nonequivalent 3-column nonnegative integer matrices with column sums n and any number of nonzero rows up to permutation of rows and columns.

Original entry on oeis.org

1, 3, 23, 162, 1338, 10585, 82694, 612700, 4337251, 29267424, 189156449, 1174258396, 7027652401, 40665520091, 228144483468, 1243885610116, 6604645458105, 34214740832064, 173212883196153, 858180731755508, 4166502761618085, 19845587224548144, 92834956645809855, 426901317803272329

Offset: 0

Andrew Howroyd, Jan 31 2020

nonn

Column k=3 of A331485.

cp's OEIS Frontend

A007717 Number of symmetric polynomial functions of degree n of a symmetric matrix (of indefinitely large size) under joint row and column permutations. Also number of multigraphs with n edges (allowing loops) on an infinite set of nodes.

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A306017 Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.

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A331315 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k columns and any number of nonzero rows with column sums n and columns in nonincreasing lexicographic order.

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A331461 Array read by antidiagonals: A(n,k) is the number of nonequivalent binary matrices with k columns and any number of nonzero rows with n ones in every column up to permutation of rows and columns.

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A316674 Number A(n,k) of paths from {0}^k to {n}^k that always move closer to {n}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A058194 Number of n-rowed matrices with entries {0,1,2,3} and all row sums 3, up to row and column permutation.

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A331721 Number of nonequivalent n-column nonnegative integer matrices with column sums 4 and any number of nonzero rows up to permutation of rows and columns.

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A331722 Number of nonequivalent 2-column nonnegative integer matrices with column sums n and any number of nonzero rows up to permutation of rows and columns.

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