A219727 -id:A219727 - cp's OEIS frontend

A318284 Number of multiset partitions of a multiset whose multiplicities are the prime indices of n.

1, 1, 2, 2, 3, 4, 5, 5, 9, 7, 7, 11, 11, 12, 16, 15, 15, 26, 22, 21, 29, 19, 30, 36, 31, 30, 66, 38, 42, 52, 56, 52, 47, 45, 57, 92, 77, 67, 77, 74, 101, 98, 135, 64, 137, 97, 176, 135, 109, 109, 118, 105, 231, 249, 97, 141, 181, 139, 297, 198, 385, 195, 269

Offset: 1

Gus Wiseman, Aug 23 2018

nonn

			The a(12) = 11 multiset partitions of {1,1,2,3}:
  {{1,1,2,3}}
  {{1},{1,2,3}}
  {{2},{1,1,3}}
  {{3},{1,1,2}}
  {{1,1},{2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
  {{2},{3},{1,1}}
  {{1},{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..700

Cf. A001055, A001970, A007716, A181821, A219727, A255906, A269134, A305936.

Cf. A317532, A318283, A318285, A318286, A318360, A318361.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[Times@@Prime/@nrmptn[n]]],{n,60}]

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}
sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i,2], j, primepi(f[i,1]))))}
count(sig)={my(n=vecsum(sig), A=O(x*x^vecmax(sig)), s=0); forpart(p=n, my(q=1/prod(i=1, #p, 1 - x^p[i] + A)); s+=prod(i=1, #sig, polcoef(q,sig[i]))*permcount(p)); s/n!}
a(n)={if(n==1, 1, my(s=sig(n)); if(#s==1, numbpart(s[1]), count(sig(n))))} \\ Andrew Howroyd, Dec 10 2018

a(n) = A001055(A181821(n)).

a(prime(n)^k) = A219727(n,k). - Andrew Howroyd, Dec 10 2018

A020555 Number of multigraphs on n labeled edges (with loops). Also number of genetically distinct states amongst n individuals.

Original entry on oeis.org

1, 2, 9, 66, 712, 10457, 198091, 4659138, 132315780, 4441561814, 173290498279, 7751828612725, 393110572846777, 22385579339430539, 1419799938299929267, 99593312799819072788, 7678949893962472351181, 647265784993486603555551, 59357523410046023899154274

Offset: 0

Gilbert Labelle (gilbert(AT)lacim.uqam.ca), Simon Plouffe, N. J. A. Sloane

nonn

Also the number of factorizations of (p_n#)^2. - David W. Wilson, Apr 30 2001
Also the number of multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018
a(n) gives the number of genetically distinct states for n diploid individuals in the case that maternal and paternal alleles transmitted to the individuals are not distinguished (if maternal and paternal alleles are distinguished, then the number of states is A000110(2n)). - Noah A Rosenberg, Aug 23 2022

			From _Gus Wiseman_, Jul 18 2018: (Start)
The a(2) = 9 multiset partitions of {1, 1, 2, 2}:
  (1122),
  (1)(122), (2)(112), (11)(22), (12)(12),
  (1)(1)(22), (1)(2)(12), (2)(2)(11),
  (1)(1)(2)(2).
(End)

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018
E. Keith Lloyd, Math. Proc. Camb. Phil. Soc., vol. 103 (1988), 277-284.
A. Murthy, Generalization of partition function, introducing Smarandache factor partitions. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Alois P. Heinz, Table of n, a(n) for n = 0..310 (first 101 terms from Vincenzo Librandi)
G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
Marko Riedel et al., Set partitions of {1,1,2,2,...,n,n}
E. A. Thompson, Gene identities and multiple relationships. Biometrics 30 (1974), 667-680. See Table 5.

Row n=2 of A219727. - Alois P. Heinz, Nov 26 2012
Cf. A007716, A007717, A014500, A014501, A020554, A094574, A316974.
See also A322764. Row 0 of the array in A322765.
Main diagonal of A346500.
Cf. A001055, A002110.

B := n -> combinat[bell](n):
P := proc(m,n) local k; global B; option remember;
if n = 0 then B(m)  else
(1/2)*( P(m+2,n-1) + P(m+1,n-1) + add( binomial(n-1,k)*P(m,k), k=0..n-1) ); fi; end;
r:=m->[seq(P(m,n),n=0..20)]; r(0); # N. J. A. Sloane, Dec 30 2018

max = 16; s = Series[Exp[-3/2 + Exp[x]/2]*Sum[Exp[Binomial[n+1, 2]*x]/n!, {n, 0, 3*max }], {x, 0, max}] // Normal; a[n_] := SeriesCoefficient[s, {x, 0, n}]*n!; Table[a[n] // Round, {n, 0, max} ] (* Jean-François Alcover, Apr 23 2014, after Vladeta Jovovic *)
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[mps[Ceiling[Range[1/2,n,1/2]]]],{n,5}] (* Gus Wiseman, Jul 18 2018 *)

Lloyd's article gives a complicated explicit formula.
E.g.f.: exp(-3/2 + exp(x)/2)*Sum_{n>=0} exp(binomial(n+1, 2)*x)/n! [probably in the Labelle paper]. - Vladeta Jovovic, Apr 27 2004
a(n) = A001055(A002110(n)^2). - Alois P. Heinz, Aug 23 2022

A346426 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 5, 4, 3, 15, 15, 11, 7, 5, 52, 52, 36, 21, 12, 7, 203, 203, 135, 74, 38, 19, 11, 877, 877, 566, 296, 141, 64, 30, 15, 4140, 4140, 2610, 1315, 592, 250, 105, 45, 22, 21147, 21147, 13082, 6393, 2752, 1098, 426, 165, 67, 30, 115975, 115975, 70631, 33645, 13960, 5317, 1940, 696, 254, 97, 42

Offset: 0

Alois P. Heinz, Jul 16 2021

Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1); A(3,1) = 7: 2*2*2*3, 2*3*4, 4*6, 2*2*6, 3*8, 2*12, 24; A(1,2) = 5: 2*3*5, 5*6, 3*10, 2*15, 30.

			A(2,2) = 11: 00|1|2, 001|2, 1|002, 0|0|1|2, 0|01|2, 0|1|02, 01|02, 00|12, 0|0|12, 0|012, 0012.
Square array A(n,k) begins:
   1,  1,   2,    5,   15,    52,    203,     877,    4140, ...
   1,  2,   5,   15,   52,   203,    877,    4140,   21147, ...
   2,  4,  11,   36,  135,   566,   2610,   13082,   70631, ...
   3,  7,  21,   74,  296,  1315,   6393,   33645,  190085, ...
   5, 12,  38,  141,  592,  2752,  13960,   76464,  448603, ...
   7, 19,  64,  250, 1098,  5317,  28009,  158926,  963913, ...
  11, 30, 105,  426, 1940,  9722,  52902,  309546, 1933171, ...
  15, 45, 165,  696, 3281, 16972,  95129,  572402, 3670878, ...
  22, 67, 254, 1106, 5372, 28582, 164528, 1015356, 6670707, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000041, A000070, A082775, A093802, A346857, A346858, A346859, A346860, A346861, A346862, A346863.
Rows n=0+1, 2-10 give: A000110, A035098, A169587, A169588, A346851, A346852, A346853, A346854, A346855, A346856.
Main diagonal gives A346424.
Antidiagonal sums give A346428.
Cf. A000079, A001055, A008277, A048993, A070826, A126442, A129306, A219727, A255903, A292508, A346520.

s:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=0,
      combinat[numbpart](n), add(b(n-j, i-1), j=0..n)))
    end:
A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

s[n_] := s[n] = Expand[If[n == 0, 1, x Sum[s[n - j] Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, PartitionsP[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
A[n_, k_] := Sum[S[k, j] b[n, j], {j, 0, k}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Aug 18 2021, after Alois P. Heinz *)

A(n,k) = A001055(A000079(n)*A070826(k+1)).
A(n,k) = Sum_{j=0..k} A048993(k,j)*A292508(n,j+1).
A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000041(n-i).

A002774 Number of bipartite partitions of n white objects and n black ones.

Original entry on oeis.org

1, 2, 9, 31, 109, 339, 1043, 2998, 8406, 22652, 59521, 151958, 379693, 927622, 2224235, 5236586, 12130780, 27669593, 62229990, 138095696, 302673029, 655627975, 1404599867, 2977831389, 6251060785, 12999299705, 26791990052, 54750235190, 110977389012

Offset: 0

N. J. A. Sloane

nonn

Number of ways to factor p^n * q^n where p and q are distinct primes.

M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 1.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, see p(n,n), page 778. - N. J. A. Sloane, Dec 30 2018
A. Murthy, Generalization of partition function, introducing Smarandache factor partitions. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
A. Murthy, Program for finding out the number of Smarandache factor partitions. (To be published in Smarandache Notions Journal).
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.14.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (terms 0..100 from Alois P. Heinz)
F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.
F. C. Auluck, On partitions of bipartite numbers, annotated scan of a few pages.
F. C. Auluck, On partitions of bipartite numbers, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 49, Issue 01, January 1953, pp. 72-83. (full article)
Katherine Ormeño Bastías, Paul Martin, and Steen Ryom-Hansen, On the spherical partition algebra, arXiv:2402.01890 [math.RT], 2024.
M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956 (Annotated scanned pages from, plus a review)
Edward Pearce-Crump, Permutation Equivariant Neural Networks for Symmetric Tensors, arXiv:2503.11276 [cs.LG], 2025. See p. 20.

Cf. A005380.
Cf. A219554. Column k=2 of A219727. - Alois P. Heinz, Nov 26 2012
Cf. also A000041, A000070, A000291, A000412, A000465, A000491, A002755, A002756, A002757, A002758, A002759, A277239.
Main diagonal of A054225 if that entry is drawn as a square array. - N. J. A. Sloane, Dec 30 2018

with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b(6^n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 27 2013

max = 26; se = Series[ Sum[ Log[1 - x^(n-k)*y^k], {n, 1, 2max}, {k, 0, n}], {x, 0, 2max}, {y, 0, 2max}]; coes = CoefficientList[ Series[ Exp[-se], {x, 0, 2max}, {y, 0, 2max}], {x, y}]; a[n_] := coes[[n+1, n+1]]; Table[a[n], {n, 0, max} ](* Jean-François Alcover, Dec 06 2011 *)

a(n) = A054225(2n, n) = A091437(2n).
a(n) ~ Zeta(3)^(19/36) * exp(3*Zeta(3)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (6*Zeta(3)^(1/3)) + Zeta'(-1) - Pi^4/(432*Zeta(3))) / (sqrt(3) * (2*Pi)^(3/2) * n^(55/36)). - Vaclav Kotesovec, Jan 30 2016
Formula (25) in the article by Auluck is incorrect. The correct formula is: p(n,n) ~ c^(19/12) * exp(3*c*n^(2/3) + 3*d*n^(1/3) + Zeta'(-1) - 3*d^2/(4*c)) / (sqrt(3) * (2*Pi)^(3/2) * n^(55/36)), where c = Zeta(3)^(1/3), d = Zeta(2)/(3*c). Also formula (24) is incorrect. - Vaclav Kotesovec, Jan 30 2016
From Vaclav Kotesovec, Feb 04 2016: (Start)
The correct formula (24) is p(m,n) ~ c^(7/4)/(2*Pi*sqrt(3)) * exp(3*c*(m*n)^(1/3) + 3*d*(m+n)/(2*(m*n)^(1/3)) - 19*log(m*n)/24 - ((m/n - 2*n/m)*log(m) + (n/m - 2*m/n)*log(n))/36 - (m/n + n/m)*(log(c)/12 + Zeta'(-1) - 1/12 + 3*d^2/(4*c)) + 3*d^2/(4*c) - 3*log(2*Pi)/4 + fi((n/m)^(1/2))),
where m and n are of the same order, c = Zeta(3)^(1/3), d = Zeta(2)/(3*c) and fi(alfa) = Integral_{t=0..infinity} (1/t)*(1/(exp(alfa*t)-1)/(exp(t/alfa)-1) - (alfa/t)/(exp(alfa*t)-1) - ((1/alfa)/t)/(exp(t/alfa)-1) + 1/t^2 + (1/4)/(exp(alfa*t)-1) + (1/4)/(exp(t/alfa)-1) - (alfa/4)/t - ((1/4)/alfa)/t).
If m = n then alfa = 1 and fi(1) = 3*Zeta'(-1) + log(2*Pi)/4 - 1/6.
For the asymptotic formula for fixed m see A054225.
(End)

Corrected using A000491.

Edited by Christian G. Bower, Jan 08 2004

A188392 T(n,k) = number of (n*k) X k binary arrays with rows in nonincreasing order and n ones in every column.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 15, 16, 4, 1, 52, 139, 39, 5, 1, 203, 1750, 862, 81, 6, 1, 877, 29388, 35775, 4079, 150, 7, 1, 4140, 624889, 2406208, 507549, 15791, 256, 8, 1, 21147, 16255738, 238773109, 127126912, 5442547, 52450, 410, 9, 1, 115975, 504717929, 32867762616

Offset: 1

R. H. Hardin, Mar 30 2011

			Array begins:
========================================================================
n\k| 1  2   3      4         5            6            7               8
---+--------------------------------------------------------------------
1  | 1  2   5     15        52          203           877           4140
2  | 1  3  16    139      1750        29388        624889       16255738
3  | 1  4  39    862     35775      2406208     238773109    32867762616
4  | 1  5  81   4079    507549    127126912   55643064708 38715666455777
5  | 1  6 150  15791   5442547   4762077620 8738543204786
6  | 1  7 256  52450  46757209 135029200594
7  | 1  8 410 154279 335279744
8  | 1  9 625 411180
9  | 1 10 915
     ...
All solutions for 6 X 2
..1..1....1..1....1..0....1..1
..1..1....1..1....1..0....1..0
..1..0....1..1....1..0....1..0
..0..1....0..0....0..1....0..1
..0..0....0..0....0..1....0..1
..0..0....0..0....0..1....0..0

Andrew Howroyd, Table of n, a(n) for n = 1..181 (terms 1..69 from R. H. Hardin)

Rows 1..8 are A000110, A020554, A165434, A165435, A165436, A165437, A188393, A188394.
Columns 3..7 are A011863(n+1), A175707, A188389, A188390, A188391.
Main diagonal gives A188388.
Cf. A188445, A219727, A330942.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); WeighT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k, q=Vec(exp(O(x*x^m) + intformal((x^n-1)/(1-x)))/(1-x))); if(n==0, 1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p,n,k), [1,n]); s*q[#q-j]))} \\ Andrew Howroyd, Dec 12 2018

A219585 Number A(n,k) of k-partite partitions of {n}^k into distinct k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 5, 2, 1, 1, 15, 40, 17, 2, 1, 1, 52, 457, 364, 46, 3, 1, 1, 203, 6995, 14595, 2897, 123, 4, 1, 1, 877, 136771, 937776, 407287, 21369, 323, 5, 1, 1, 4140, 3299218, 88507276, 107652681, 10200931, 148257, 809, 6, 1

Offset: 0

Alois P. Heinz, Nov 23 2012

A(n,k) is the number of factorizations of m^n into distinct factors where m is a product of k distinct primes. A(2,2) = 5: (2*3)^2 = 36 has 5 factorizations into distinct factors: 36, 3*12, 4*9, 2*18, 2*3*6.

			A(1,3) = 5: [(1,1,1)], [(1,1,0),(0,0,1)], [(1,0,1),(0,1,0)], [(1,0,0),(0,1,0),(0,0,1)], [(0,1,1),(1,0,0)].
A(3,2) = 17: [(3,3)], [(3,0),(0,3)], [(3,2),(0,1)], [(2,3),(1,0)], [(3,1),(0,2)], [(2,2),(1,1)], [(1,3),(2,0)], [(2,1),(1,2)], [(2,1),(1,1),(0,1)], [(3,0),(0,2),(0,1)], [(2,2),(1,0),(0,1)], [(2,1),(0,2),(1,0)], [(1,2),(2,0),(0,1)], [(1,2),(1,1),(1,0)], [(0,3),(2,0),(1,0)], [(2,0),(1,1),(0,2)], [(2,0),(0,2),(1,0),(0,1)].
Square array A(n,k) begins:
  1,  1,   1,      1,          1,            1,         1, ...
  1,  1,   2,      5,         15,           52,       203, ...
  1,  1,   5,     40,        457,         6995,    136771, ...
  1,  2,  17,    364,      14595,       937776,  88507276, ...
  1,  2,  46,   2897,     407287,    107652681,  ...
  1,  3, 123,  21369,   10200931,  10781201973,  ...
  1,  4, 323, 148257,  233051939,  ...
  1,  5, 809, 970246, 4909342744,  ...

Andrew Howroyd, Table of n, a(n) for n = 0..209

Columns k=0..5 give: A000012, A000009, A219554, A219560, A219561, A219565.
Rows n=0..3 give: A000012, A000110, A094574, A319591.
Cf. A188445, A219727 (partitions of {n}^k into k-tuples), A318286.

f[n_, k_] := f[n, k] = 1/2 Product[Sum[O[x[j]]^(n+1), {j, 1, k}]+1+ Product[x[j]^i[j], {j, 1, k}], Evaluate[Sequence @@ Table[{i[j], 0, n}, {j, 1, k}]]];
a[0, ] = a[, 0] = 1; a[n_, k_] := SeriesCoefficient[f[n, k], Sequence @@ Table[{x[j], 0, n}, {j, 1, k}]];
Table[Print[a[n-k, k]]; a[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013, updated Sep 16 2019 *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); EulerT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)} \\ Andrew Howroyd, Dec 16 2018

A(n,k) = [(Product_{j=1..k} x_j)^n] 1/2 * Product_{i_1,...,i_k>=0} (1+Product_{j=1..k} x_j^i_j).

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).

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

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).

A331161 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of nonzero rows with column sums n and rows in nonincreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 7, 3, 1, 0, 1, 43, 28, 5, 1, 0, 1, 403, 599, 104, 7, 1, 0, 1, 5245, 23243, 6404, 332, 11, 1, 0, 1, 89132, 1440532, 872681, 57613, 1032, 15, 1, 0, 1, 1898630, 131530132, 222686668, 26560747, 473674, 2983, 22, 1

Offset: 0

Andrew Howroyd, Jan 10 2020

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

			Array begins:
====================================================================
n\k | 0  1    2       3           4             5              6
----+---------------------------------------------------------------
  0 | 1  1    0       0           0             0              0 ...
  1 | 1  1    1       1           1             1              1 ...
  2 | 1  2    7      43         403          5245          89132 ...
  3 | 1  3   28     599       23243       1440532      131530132 ...
  4 | 1  5  104    6404      872681     222686668    95605470805 ...
  5 | 1  7  332   57613    26560747   26852940027 52296207431182 ...
  6 | 1 11 1032  473674   712725249 2776638423133 ...
  7 | 1 15 2983 3599384 17328777789 ...
  ...
The A(2,2) = 7 matrices are:
   [2 1]   [2 0]   [1 2]   [1 1]   [2 0]   [1 0]   [1 0]
   [0 1]   [0 2]   [1 0]   [1 0]   [0 1]   [1 0]   [1 0]
                           [0 1]   [0 1]   [0 2]   [0 1]
                                                   [0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..209

Rows n=1..3 are A000012, A014501, A331196.
Columns k=0..2 are A000012, A000041, A331197.
Cf. A219727, A330158, A331039, A331126, A331160, A331315.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(EulerT(v)[n], k)*k!/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k, q=Vec(exp(O(x*x^m) + intformal((x^n-1)/(1-x)))/(1-x))); if(n==0, k<=1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p, n, k), [1, n]); s*q[#q-j]))}

A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A219727(n, j).

A330158(n) = Sum_{d|n} A(n/d, d).

cp's OEIS Frontend

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A346426 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A002774 Number of bipartite partitions of n white objects and n black ones.

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A188392 T(n,k) = number of (n*k) X k binary arrays with rows in nonincreasing order and n ones in every column.

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A219585 Number A(n,k) of k-partite partitions of {n}^k into distinct k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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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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