A331461 -id:A331461 - cp's OEIS frontend

A262809 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 13, 13, 1, 1, 1, 75, 409, 63, 1, 1, 1, 541, 23917, 16081, 321, 1, 1, 1, 4683, 2244361, 10681263, 699121, 1683, 1, 1, 1, 47293, 308682013, 14638956721, 5552351121, 32193253, 8989, 1, 1, 1, 545835, 58514835289, 35941784497263, 117029959485121, 3147728203035, 1538743249, 48639, 1, 1

Offset: 0

Alois P. Heinz, Oct 02 2015

Also, A(n,k) is the number of alignments for k sequences of length n each (Slowinski 1998).
Row r > 0 is asymptotic to sqrt(r*Pi) * (r^(r-1)/(r-1)!)^n * n^(r*n+1/2) / (2^(r/2) * exp(r*n) * (log(2))^(r*n+1)), or equivalently to sqrt(r) * (r^(r-1)/(r-1)!)^n * (n!)^r / (2^r * (Pi*n)^((r-1)/2) * (log(2))^(r*n+1)). - Vaclav Kotesovec, Mar 23 2016
From Vaclav Kotesovec, Mar 23 2016: (Start)
Column k > 0 is asymptotic to sqrt(c(k)) * d(k)^n / (Pi*n)^((k-1)/2), where c(k) and d(k) are roots of polynomial equations of degree k, independent on n.
---------------------------------------------------
k               d(k)
---------------------------------------------------
2              5.8284271247461900976033774484193...
3             56.9476283720414911685286267804411...
4            780.2794068067951456595241495989622...
5          13755.2719024115081712083954421541320...
6         296476.9162644200814909862281498491264...
7        7553550.6198338218721069097516499501996...
8      222082591.6017202421029000117685530884167...
9     7400694480.0494436216324852038000444393262...
10  275651917450.6709238286995776605620357737005...
---------------------------------------------------
d(k) is a root of polynomial:
---------------------------------------------------
k=2, 1 - 6*d + d^2
k=3, -1 + 3*d - 57*d^2 + d^3
k=4, 1 - 12*d - 218*d^2 - 780*d^3 + d^4
k=5, -1 + 5*d - 1260*d^2 - 3740*d^3 - 13755*d^4 + d^5
k=6, 1 - 18*d - 5397*d^2 - 123696*d^3 + 321303*d^4 - 296478*d^5 + d^6
k=7, -1 + 7*d - 24031*d^2 - 374521*d^3 - 24850385*d^4 + 17978709*d^5 - 7553553*d^6 + d^7
k=8, 1 - 24*d - 102692*d^2 - 9298344*d^3 + 536208070*d^4 - 7106080680*d^5 - 1688209700*d^6 - 222082584*d^7 + d^8
(End)
d(k) = (2^(1/k) - 1)^(-k). - David Bevan, Apr 07 2022
d(k) is asymptotic to (k/log(2))^k/sqrt(2). - David Bevan, Apr 07 2022
A(n,k) is the number of binary matrices with k columns and any number of nonzero rows with n ones in every column. - Andrew Howroyd, Jan 23 2020

			A(2,2) = 13: [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,0)], [(2,2),(1,2),(1,1),(1,0),(0,0)], [(2,2),(2,1),(1,1),(0,1),(0,0)], [(2,2),(2,1),(1,1),(0,0)], [(2,2),(2,1),(1,1),(1,0),(0,0)], [(2,2),(2,1),(2,0),(0,1),(0,0)], [(2,2),(2,1),(1,0),(0,0)], [(2,2),(1,1),(0,1),(0,0)], [(2,2),(1,1),(0,0)], [(2,2),(1,1),(1,0),(0,0)].
Square array A(n,k) begins:
  1, 1,    1,        1,             1,                   1, ...
  1, 1,    3,       13,            75,                 541, ...
  1, 1,   13,      409,         23917,             2244361, ...
  1, 1,   63,    16081,      10681263,         14638956721, ...
  1, 1,  321,   699121,    5552351121,     117029959485121, ...
  1, 1, 1683, 32193253, 3147728203035, 1050740615666453461, ...

Alois P. Heinz, Antidiagonals n = 0..48, flattened
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522

Columns: A000012 (k=0 and k=1), A001850 (k=2), A126086 (k=3), A263064 (k=4), A263065 (k=5), A263066 (k=6), A263067 (k=7), A263068 (k=8), A263069 (k=9), A263070 (k=10).
Rows: A000012 (n=0), A000670 (n=1), A055203 (n=2), A062208 (n=3), A062205 (n=4), A263061 (n=5), A263062 (n=6), A062204 (n=7), A263063 (n=8), A263071 (n=9), A263072 (n=10).
Main diagonal: A262810.
Cf. A210472, A225094, A227578, A227655, A229142, A229345, A263159, A316674.
Cf. A188392, A330942, A331461, A331637.

A:= (n, k)-> 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[, 0] =  1; A[n, k_] := Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^k, {i, 0, j}], {j, 0, k*n}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jul 22 2016, after Alois P. Heinz *)

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

A(n,k) = Sum_{j=0..k*n} Sum_{i=0..j} (-1)^i*C(j,i)*C(j-i,n)^k.
A(n,k) = Sum_{i >= 0} binomial(i,n)^k/2^(i+1). - Peter Bala, Jan 30 2018
A(n,k) = Sum_{j=0..n*k} binomial(j,n)^k * Sum_{i=j..n*k} (-1)^(i-j) * binomial(i,j). - Andrew Howroyd, Jan 23 2020

A330942 Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of nonzero rows with n ones in every column and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 7, 1, 1, 1, 8, 75, 32, 1, 1, 1, 16, 1105, 2712, 161, 1, 1, 1, 32, 20821, 449102, 116681, 842, 1, 1, 1, 64, 478439, 122886128, 231522891, 5366384, 4495, 1, 1, 1, 128, 12977815, 50225389432, 975712562347, 131163390878, 256461703, 24320, 1, 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 labeled n-uniform hypergraphs with multiple edges allowed and with k edges and no isolated vertices. When n=2 these objects are multigraphs.

			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 1    7        75           1105            20821 ...
  3 | 1 1   32      2712         449102        122886128 ...
  4 | 1 1  161    116681      231522891     975712562347 ...
  5 | 1 1  842   5366384   131163390878 8756434117294432 ...
  6 | 1 1 4495 256461703 78650129124911 ...
  ...
The A(2,2) = 7 matrices are:
   [1 0]  [1 0]  [1 0]  [1 1]  [1 0]  [1 0]  [1 1]
   [1 0]  [0 1]  [0 1]  [1 0]  [1 1]  [0 1]  [1 1]
   [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [1 1]
   [0 1]  [0 1]  [1 0]

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

Rows n=1..3 are A000012, A121316, A136246.
Columns k=0..3 are A000012, A000012, A226994, A137220.
The version with nonnegative integer entries is A331315.
Other variations considering distinct rows and columns and equivalence under different combinations of permutations of rows and columns are:
All solutions: A262809 (all), A331567 (distinct rows).
Up to row permutation: A188392, A188445, A331126, A331039.
Up to column permutation: this sequence, A331571, A331277, A331569.
Nonisomorphic: A331461, A331510, A331508, A331509.
Cf. A331638.

T[n_, k_] := With[{m = n k}, Sum[Binomial[Binomial[j, n] + k - 1, k] Sum[ (-1)^(i - j) Binomial[i, j], {i, j, m}], {j, 0, m}]];
Table[T[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 10 2020, from PARI *)

T(n, k)={my(m=n*k); sum(j=0, m, binomial(binomial(j, 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)+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))*A262809(n, j)/k!.
A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331277(n, j).
A331638(n) = Sum_{d|n} A(n/d, d).

A050913 Pure 2-complexes on an infinite set of nodes with n multiple 2-simplexes. Also n-rowed binary matrices with all row sums 3, up to row and column permutation.

Original entry on oeis.org

1, 1, 4, 16, 93, 652, 6369, 79568, 1256425, 24058631, 543204998, 14138916124, 417362929209, 13798729189578, 505990335048034, 20415765544541866, 900364519682003919, 43155049922002494115, 2236988329443856718604, 124862936181977439454012, 7476052709321753156375756, 478506183522725779096476581, 32638841238874891261354722405, 2365895836144423508306322639848, 181785988254681334224483607437510, 14771116583797935886529061991645404, 1266545494725474774697216198539818982

Offset: 0

Vladeta Jovovic, Dec 29 1999

nonn

Vladeta Jovovic, Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation

Row n=3 of A331461.

Cf. A050911, A050912, A001400, A014395, A082789, A082790, A058790.

More terms from T. Forbes (anthony.d.forbes(AT)googlemail.com), May 24 2003

A331508 Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set multipartitions (multisets of sets) on a k-set.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 5, 3, 1, 1, 0, 1, 11, 12, 4, 1, 1, 0, 1, 26, 66, 25, 5, 1, 1, 0, 1, 68, 445, 278, 44, 6, 1, 1, 0, 1, 177, 4279, 5532, 966, 73, 7, 1, 1, 0, 1, 497, 53340, 200589, 53535, 2957, 112, 8, 1, 1, 0, 1, 1476, 846254, 11662671, 7043925, 431805, 8149, 166, 9, 1, 1

Offset: 0

Andrew Howroyd, Jan 18 2020

An n-regular set multipartition is a finite multiset of nonempty sets in which each element appears in n blocks.
A set multipartition is T_0 if for every two distinct elements there exists a block containing one but not the other element.
A(n,k) is the number of nonequivalent binary matrices with k distinct columns and any number of nonzero rows with n ones in every column up to permutation of rows and columns.
A(n,k) is the number of non-isomorphic set-systems with k parts each of size n.

			Array begins:
===============================================
n\k | 0 1 2  3    4      5       6        7
----+------------------------------------------
  0 | 1 1 0  0    0      0       0        0 ...
  1 | 1 1 1  1    1      1       1        1 ...
  2 | 1 1 2  5   11     26      68      177 ...
  3 | 1 1 3 12   66    445    4279    53340 ...
  4 | 1 1 4 25  278   5532  200589 11662671 ...
  5 | 1 1 5 44  966  53535 7043925 ...
  6 | 1 1 6 73 2957 431805 ...
  ...
The A(2,3) = 5 matrices are:
  [1 0 0]  [1 1 0]  [1 1 1]  [1 1 0]  [1 1 0]
  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 1]  [1 0 1]
  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 1]
  [0 1 0]  [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 (first 17 antidiagonals)

Rows n=1..6 are A000012, A000664, A058790, A058791, A058792, A058793.

Cf. A283877, A306019, A330942, A331126, A331277, A331461, A331509, A331510.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(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)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))[k]}
T(n,k)={my(m=n*k, s=0); if(m==0, k<=1, forpart(q=m, my(g=sum(t=1, k, K(q, t, n)*x^t/t) + O(x*x^k)); s+=permcount(q)*polcoef(exp(g - subst(g,x,x^2)), k)); s/m!)}
{ for(n=0, 6, for(k=0, 5, print1(T(n, k), ", ")); print) } \\ Andrew Howroyd, Jan 16 2024

A306019(n) = Sum_{d|n} A(n/d, d).

A306018 Number of non-isomorphic set multipartitions of weight n in which all parts have the same size.

Original entry on oeis.org

1, 1, 3, 4, 9, 8, 24, 16, 51, 47, 115, 57, 420, 102, 830, 879, 2962, 298, 15527, 491, 41275, 80481, 133292, 1256, 2038182, 58671, 2386862, 24061887, 23570088, 4566, 600731285, 6843, 1303320380, 14138926716, 1182784693, 1820343112, 542834549721, 21638, 31525806080

Offset: 0

Gus Wiseman, Jun 17 2018

nonn

A set multipartition of weight n is a finite multiset of finite nonempty sets whose cardinalities sum to n.

Number of distinct binary matrices with all row sums equal and total sum n, up to row and column permutations. - Andrew Howroyd, Sep 05 2018

			Non-isomorphic representatives of the a(6) = 24 set multipartitions in which all parts have the same size:
{{1,2,3,4,5,6}}
{{1,2,3},{1,2,3}}
{{1,2,3},{4,5,6}}
{{1,2,5},{3,4,5}}
{{1,3,4},{2,3,4}}
{{1,2},{1,2},{1,2}}
{{1,2},{1,3},{2,3}}
{{1,2},{3,4},{3,4}}
{{1,2},{3,4},{5,6}}
{{1,2},{3,5},{4,5}}
{{1,3},{2,3},{2,3}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}
{{1},{1},{1},{1},{1},{1}}
{{1},{1},{1},{2},{2},{2}}
{{1},{1},{2},{2},{2},{2}}
{{1},{1},{2},{2},{3},{3}}
{{1},{2},{2},{2},{2},{2}}
{{1},{2},{2},{3},{3},{3}}
{{1},{2},{3},{3},{3},{3}}
{{1},{2},{3},{3},{4},{4}}
{{1},{2},{3},{4},{4},{4}}
{{1},{2},{3},{4},{5},{5}}
{{1},{2},{3},{4},{5},{6}}

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

Cf. A000005, A000041, A001315, A007716, A038041, A049311, A283877, A298422, A304942, A306017, A306019, A306020, A306021, A331461.

\\ See A304942 for Blocks
a(n)={sumdiv(n, d, Blocks(n/d, n, d))} \\ Andrew Howroyd, Sep 05 2018

a(p) = A000041(p) + 1 for prime p. - Andrew Howroyd, Sep 06 2018

a(n) = Sum_{d|n} A331461(n/d, d). - Andrew Howroyd, Feb 09 2020

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

A058783 Number of n-rowed binary matrices with all row sums 4, up to row and column permutation; bipartite graphs with all nodes of degree 4 in a distinguished bipartite n-block, up to isomorphism.

Original entry on oeis.org

1, 1, 5, 30, 332, 6414, 226041, 12848128, 1064292052, 119252205304, 17239075745190, 3113843614322530, 686491853010870156, 181386885029173595218, 56595769613704915215101, 20597536264356706999502705, 8652615103513713632218678155, 4157673185369121151936091009448

Offset: 0

Vladeta Jovovic, Nov 28 2000

nonn

Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation

Row n=4 of A331461.

Cf. A050535, A050913, A058784, A058785.

Terms a(8)-a(17) from Max Alekseyev, May 04 2018

A331485 Array read by antidiagonals: A(n,k) is the number of nonequivalent nonnegative integer matrices with k columns and any number of nonzero rows with column sums n up to permutation of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 3, 1, 1, 5, 23, 21, 5, 1, 1, 7, 79, 162, 66, 7, 1, 1, 11, 274, 1636, 1338, 192, 11, 1, 1, 15, 1003, 19977, 43686, 10585, 565, 15, 1, 1, 22, 3763, 298416, 2142277, 1178221, 82694, 1579, 22, 1, 1, 30, 14723, 5300296, 149056260, 232984145, 30370346, 612700, 4348, 30, 1

Offset: 0

Andrew Howroyd, Jan 18 2020

A(n,k) is the number of non-isomorphic multiset partitions (multisets of multisets) with k parts each of size n.

			Array begins:
============================================================
n\k | 0  1   2     3        4           5              6
----+-------------------------------------------------------
  0 | 1  1   1     1        1           1              1 ...
  1 | 1  1   2     3        5           7             11 ...
  2 | 1  2   7    23       79         274           1003 ...
  3 | 1  3  21   162     1636       19977         298416 ...
  4 | 1  5  66  1338    43686     2142277      149056260 ...
  5 | 1  7 192 10585  1178221   232984145    74676589469 ...
  6 | 1 11 565 82694 30370346 23412296767 33463656939910 ...
  ...
The A(2,2) = 7 matrices are:
  [1 0]  [2 0]  [1 1]  [2 1]  [2 0]  [1 1]  [2 2]
  [1 0]  [0 1]  [1 0]  [0 1]  [0 2]  [1 1]
  [0 1]  [0 1]  [0 1]
  [0 1]

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

Rows n=0..4 are A000012, A000041, A007717, A058194, A331721.
Columns k=0..3 are A000012, A000041, A331722, A331723.
Cf. A219727, A306017, A316674, A318951, A331315, A331461.

\\ See A318951 for RowSumMats
T(n, k)={RowSumMats(k, n*k, n)}
{ for(n=0, 7, for(k=0, 6, print1(T(n, k), ", ")); print) }

A306017(n) = Sum_{d|n} A(n/d, d).

A058784 Number of n-rowed binary matrices with all row sums 5, up to row and column permutation; bipartite graphs with all nodes of degree 5 in a distinguished bipartite n-block, up to isomorphism.

Original entry on oeis.org

1, 1, 6, 50, 1062, 56712, 7295812, 1817321457, 750572034647, 467477187494249, 413492386704135759, 498000976932085045800, 791133947048635010571251, 1616618616930983136958492360, 4162734189568898641317813946712, 13276910075946265356264268787123401

Offset: 0

Vladeta Jovovic, Nov 28 2000

nonn

Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation

Row n=5 of A331461.

Cf. A050535, A050913, A058783, A058785.

a(7)-a(15) from Max Alekseyev, May 04 2018

A058785 Number of n-rowed binary matrices with all row sums 6, up to row and column permutation; bipartite graphs with all nodes of degree 6 in a distinguished bipartite n-block, up to isomorphism.

Original entry on oeis.org

1, 1, 7, 80, 3117, 441881, 195486906, 200065951078, 390629444879796, 1295710020278986959, 6764950163433890601997, 52637015698577710285832949, 585529138290221945767868247037, 9010635399744213671095287436920755

Offset: 0

Vladeta Jovovic, Nov 28 2000

nonn

Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation

Row n=6 of A331461.

Cf. A050535, A050913, A058783, A058784.

a(6)-a(13) from Max Alekseyev, May 04 2018

A331510 Array read by antidiagonals: A(n,k) is the number of nonequivalent binary matrices with k columns and any number of distinct 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, 0, 1, 1, 3, 1, 0, 1, 1, 5, 4, 0, 0, 1, 1, 7, 12, 3, 0, 0, 1, 1, 11, 36, 23, 1, 0, 0, 1, 1, 15, 124, 191, 30, 0, 0, 0, 1, 1, 22, 412, 2203, 837, 23, 0, 0, 0, 1, 1, 30, 1500, 31313, 41664, 2688, 12, 0, 0, 0, 1

Offset: 0

Andrew Howroyd, Jan 18 2020

			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 0 1 4 12  36  124 412 ...
  3 | 1 0 0 3 23 191 2203 ...
  4 | 1 0 0 1 30 837 ...
  5 | 1 0 0 0 23 ...
  ...
The A(2,3) = 4 matrices are:
  [1 1 1]  [1 1 0]  [1 1 1]  [1 1 0]
  [1 0 0]  [1 0 1]  [1 1 0]  [1 0 1]
  [0 1 0]  [0 1 0]  [0 0 1]  [0 1 1]
  [0 0 1]  [0 0 1]

Rows n=1..3 are A000041, A331717, A331718.
Column k=5 is A331719.
Cf. A188445, A330942, A331461, A331508, A331509.

A(n,k) = 0 for k > 0, n > 2^(k-1).

A(n,k) = A(2^(k-1) - n, k) for k > 0, n <= 2^(k-1).

a(58)-a(65) from Andrew Howroyd, Feb 08 2020

cp's OEIS Frontend

A262809 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A330942 Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of nonzero rows with n ones in every column and columns in nonincreasing lexicographic order.

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A050913 Pure 2-complexes on an infinite set of nodes with n multiple 2-simplexes. Also n-rowed binary matrices with all row sums 3, up to row and column permutation.

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A331508 Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set multipartitions (multisets of sets) on a k-set.

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A306018 Number of non-isomorphic set multipartitions of weight n in which all parts have the same size.

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A058783 Number of n-rowed binary matrices with all row sums 4, up to row and column permutation; bipartite graphs with all nodes of degree 4 in a distinguished bipartite n-block, up to isomorphism.

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A331485 Array read by antidiagonals: A(n,k) is the number of nonequivalent nonnegative integer matrices with k columns and any number of nonzero rows with column sums n up to permutation of rows and columns.

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A058784 Number of n-rowed binary matrices with all row sums 5, up to row and column permutation; bipartite graphs with all nodes of degree 5 in a distinguished bipartite n-block, up to isomorphism.

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