A331039 -id:A331039 - cp's OEIS frontend

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.

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).

A060053 Number of r-bicoverings (or restricted proper 2-covers) of an n-set.

Original entry on oeis.org

1, 0, 1, 5, 43, 518, 8186, 163356, 3988342, 116396952, 3985947805, 157783127673, 7131072006829, 364166073164914, 20827961078794845, 1323968417981743817, 92917890994442697487, 7157607311779373890120, 602043767970637640566684

Offset: 0

Vladeta Jovovic, Feb 15 2001

A bicovering is called an r-bicovering if the intersection of every two blocks contains at most one element.
Another name for this sequence is the number of restricted proper 2-covers of [1,...,n].
Number of T_0 2-regular set-systems on an n-set. - Andrew Howroyd, Jan 08 2020

			There are 5 r-bicoverings of a 3-set: 1 3-block bicovering {{1, 2}, {1, 3}, {2, 3}} and 4 4-block bicoverings {{1}, {2}, {3}, {1, 2, 3}}, {{2}, {3}, {1, 2}, {1, 3}}, {{1}, {3}, {1, 2}, {2, 3}}, {{1}, {2}, {1, 3}, {2, 3}}.
G.f. = 1 + x^2 + 5*x^3 + 43*x^4 + 518*x^5 + 8186*x^6 + 163356*x^7 + ...

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983. (See p. 203.)

Robert Gerbicz, Table of n, a(n) for n = 0..100
Peter Cameron, Thomas Prellberg, and Dudley Stark, Asymptotic enumeration of 2-covers and line graphs, arXiv:0707.0664 [math.CO], 2007.
Peter Cameron, Thomas Prellberg, and Dudley Stark, Asymptotic enumeration of 2-covers and line graphs, Discrete Math. 310 (2010), no. 2, 230-240. (See v_n.)

Row 2 of A331039.
Row sums of A060052.
Cf. A060051, A002718, A059443, A003462, A059945-A059951.

A060053 := proc(n) local h, m; h := (m,n) -> add((-1/2)^k*binomial(m*(m-1)/2,n-k)/k!, k=0..n); n!*add(h(m,n)/m!, m=0..3*n); ceil(evalf(%/exp(1),99)) end: seq(A060053(i), i=0..18);
# Caveat computator! Limited accuracy. Do not use it for n > 50. - Peter Luschny, Jul 06 2011

f[n_] := FullSimplify[(n!/E)*Sum[(1/m!)*Sum[(-1/2)^k*Binomial[m*(m - 1)/2,
n - k]/k!, {k, 0, n}], {m, 0, Infinity}]] (* Robert G. Wilson v, Jul 03 2011 *)

a(n)=round(n!/exp(1)*sum(m=0,3*n+1,1/m!*sum(k=0,n,(-1/2)^k*binomial(m*(m-1)/2,n-k)/k!)))

\\ here egf1 is A020556 as e.g.f.
egf1(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, (-1)^k*binomial(i,k)*polcoef(bell, 2*i-k))*x^i/i!) + O(x*x^n)}
seq(n)={my(A=egf1(n), B=log(1+x + O(x*x^n))/2); Vec(serlaplace(exp(-x/2 + O(x*x^n))*sum(k=0, n, polcoef(A,k)*B^k)))} \\ Andrew Howroyd, Jan 13 2020

E.g.f. for number of k-block r-bicoverings of an n-set is exp(-x-x^2*y/2)*Sum_{i=0..inf} (1+y)^binomial(i, 2)*x^i/i!.
a(n) = row sums of A060052.
Inverse binomial transform of A014500. - Vladeta Jovovic, Aug 22 2006
The e.g.f.'s of A002718 (T(x)) and A060053 (V(x)) are related by T(x) = V(e^x-1).
The e.g.f.'s of A014500 (U(x)) and A060053 (V(x)) are related by U(x) = e^x*V(x).
E.g.f.: exp(-x/2)*(Sum_{k>=0} A020556(k)*(log(1 + x)/2)^k/k!). - Andrew Howroyd, Jan 13 2020

A188445 T(n,k) is the number of (n*k) X k binary arrays with nonzero rows in decreasing order and n ones in every column.

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 15, 8, 0, 0, 52, 80, 5, 0, 0, 203, 1088, 205, 1, 0, 0, 877, 19232, 11301, 278, 0, 0, 0, 4140, 424400, 904580, 67198, 205, 0, 0, 0, 21147, 11361786, 101173251, 24537905, 250735, 80, 0, 0, 0, 115975, 361058000, 15207243828, 13744869502

Offset: 1

R. H. Hardin, Mar 31 2011

			Array begins:
============================================================================
n\k| 1 2 3   4       5          6             7              8             9
---+------------------------------------------------------------------------
1  | 1 2 5  15      52        203           877           4140         21147
2  | 0 1 8  80    1088      19232        424400       11361786     361058000
3  | 0 0 5 205   11301     904580     101173251    15207243828 2975725761202
4  | 0 0 1 278   67198   24537905   13744869502 11385203921707 ...
5  | 0 0 0 205  250735  425677958 1184910460297 ...
6  | 0 0 0  80  621348 5064948309 ...
7  | 0 0 0  15 1058139 ...
8  | 0 0 0   1 ...
...
Some solutions for 16 X 4:
  1 1 1 0    1 1 1 1    1 1 1 1    1 1 1 0    1 1 1 1
  1 0 1 1    1 1 0 1    1 1 0 0    1 0 1 1    1 1 0 0
  1 0 1 0    1 0 1 1    1 0 1 1    1 0 0 1    1 0 1 1
  1 0 0 1    1 0 0 0    1 0 0 0    1 0 0 0    1 0 0 0
  0 1 1 1    0 1 1 0    0 1 1 1    0 1 1 0    0 1 1 1
  0 1 0 1    0 1 0 0    0 1 0 0    0 1 0 1    0 1 0 0
  0 1 0 0    0 0 1 1    0 0 1 1    0 1 0 0    0 0 1 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 1 1    0 0 0 1
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0

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

Rows 1..5 are A000110, A002718, A060486, A188446, A188447.
Columns 5..6 are A331127, A331129.
Column sums are A319190.
Cf. A059443, A060487, A188392, A219585, A318361, A330942, A330964, A331039.

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+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) = 0 for n > 2^(k-1). - Andrew Howroyd, Jan 24 2020

A060070 Number of T_0-tricoverings of an n-set.

Original entry on oeis.org

1, 0, 0, 5, 175, 9426, 751365, 84012191, 12644839585, 2479642897109, 617049443550205, 190678639438170502, 71860665148118443795, 32527628234581386962713, 17454341903042193018433239, 10978059489008346809004564072, 8013452442154510131205645967978

Offset: 0

Vladeta Jovovic, Feb 21 2001

nonn

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering. A covering of a set is a T_0-covering if for every two distinct elements of the set there exists a block of the covering containing one but not the other element.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Vladeta Jovovic, T_0-tricoverings of a 4-set

Row n=3 of A331039.
Row sums of A059530.
Cf. A060051, A060053, A060069, A060486.

seq(n)={my(m=2*n, y='y + O('y^(n+1))); Vec(serlaplace(subst(Pol(exp(-x + x^2/2 + x^3*y/3 + O(x*x^m))*sum(k=0, m, (1+y)^binomial(k, 3)*exp(-x^2*(1+y)^k/2 + O(x*x^m))*x^k/k!)), x, 1)))} \\ Andrew Howroyd, Jan 30 2020

E.g.f. for k-block T_0-tricoverings of an n-set is exp(-x+1/2*x^2+1/3*x^3*y)*Sum_{i=0..inf}(1+y)^binomial(i, 3)*exp(-1/2*x^2*(1+y)^i)*x^i/i!.

a(n) = Sum_{k=0..n} Stirling1(n, k)*A060486(k). - Andrew Howroyd, Jan 08 2020

Terms a(15) and beyond from Andrew Howroyd, Jan 08 2020

A060052 Triangle read by rows: T(n,k) gives number of r-bicoverings of an n-set with k blocks, n >= 2, k = 3..n+floor(n/2).

Original entry on oeis.org

1, 1, 4, 0, 15, 25, 3, 0, 30, 222, 226, 40, 0, 30, 1230, 3670, 2706, 535, 15, 0, 0, 5040, 39900, 69450, 40405, 8141, 420, 0, 0, 15120, 345240, 1254960, 1498035, 722275, 142877, 9730, 105, 0, 0, 30240, 2492280, 18587520, 40701780, 36450820, 15031204, 2871240, 226828, 5040

Offset: 2

Vladeta Jovovic, Feb 15 2001

A bicovering is r-bicovering if intersection of every two blocks contains at most one element.

			Triangle starts:
[1],
[1, 4],
[0, 15, 25, 3],
[0, 30, 222, 226, 40],
[0, 30, 1230, 3670, 2706, 535, 15],
[0, 0, 5040, 39900, 69450, 40405, 8141, 420],
[0, 0, 15120, 345240, 1254960, 1498035, 722275, 142877, 9730, 105],
[0, 0, 30240, 2492280, 18587520, 40701780, 36450820, 15031204, 2871240, 226828, 5040],
...

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Andrew Howroyd, Table of n, a(n) for n = 2..1802 (rows n=2..50)
Table

Row sums are A060053.
Column sums are A060051.
Cf. A059443, A059530, A060092, A060487, A060492, A276640, A331039.

\\ returns k-th column as vector.
C(k)=if(k<3, [], Vecrev(serlaplace(polcoef(exp(-x-1/2*x^2*y + O(x*x^k))*sum(i=0, 3*k\2, (1+y)^binomial(i, 2)*x^i/i!), k))/y)) \\ Andrew Howroyd, Jan 30 2020

T(n)={my(m=(3*n\2), y='y + O('y^(n+1))); my(g=exp(-x-1/2*x^2*y + O(x*x^m))*sum(k=0, m, (1+y)^binomial(k, 2)*x^k/k!)); Mat([Col(serlaplace(p), -n) | p<-Vec(g)[2..m+1]])}
{ my(A=T(8)); for(n=2, matsize(A)[1], print(A[n, 3..3*n\2])) } \\ Andrew Howroyd, Jan 30 2020

E.g.f.: A(x, y) = exp(-x-1/2*x^2*y)*Sum_{i>=0} (1+y)^binomial(i, 2)*x^i/i!.
T(n, k) = (n!/k!) * A276640(k, n). - David Pasino, Sep 22 2016
T(n,k) = 0 for n > binomial(k,2). - Andrew Howroyd, Jan 30 2020

Zeros inserted into data by Andrew Howroyd, Jan 30 2020

A331569 Array read by antidiagonals: A(n,k) is the number of binary matrices with k distinct columns and any number of distinct nonzero rows with n ones in every column and columns in decreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 3, 0, 1, 0, 1, 17, 0, 0, 1, 0, 1, 230, 184, 0, 0, 1, 0, 1, 3264, 16936, 840, 0, 0, 1, 0, 1, 60338, 2711904, 768785, 0, 0, 0, 1, 0, 1, 1287062, 675457000, 1493786233, 21770070, 0, 0, 0, 1, 0, 1, 31900620, 232383728378, 5254074934990, 585810653616, 328149360, 0, 0, 0, 1

Offset: 0

Andrew Howroyd, Jan 20 2020

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

A(n,k) is the number of k-block n-uniform T_0 set systems without isolated vertices.

			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 0 3  17       230            3264             60338 ...
  3 | 1 0 0 184     16936         2711904         675457000 ...
  4 | 1 0 0 840    768785      1493786233     5254074934990 ...
  5 | 1 0 0   0  21770070    585810653616 30604798810581906 ...
  6 | 1 0 0   0 328149360 161087473081920 ...
  ...
The A(2,2) = 3 matrices are:
   [1 1]  [1 0]  [1 0]
   [1 0]  [1 1]  [0 1]
   [0 1]  [0 1]  [1 1]

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

Rows n=1..4 are A000012, A331649, A094631, A331650.

Cf. A330942, A331039, A331567, A331570, A331571, A331572, A331651.

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]]++); binomial(WeighT(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)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, k<=1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }

A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A331567(n, j)/k!.
A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331571(n, j).
A331651(n) = Sum_{d|n} A(n/d, d).

A331126 Array read by antidiagonals: A(n,k) is the number of 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, 9, 3, 1, 1, 0, 1, 70, 29, 4, 1, 1, 0, 1, 794, 666, 68, 5, 1, 1, 0, 1, 12055, 28344, 3642, 134, 6, 1, 1, 0, 1, 233238, 1935054, 469368, 14951, 237, 7, 1, 1, 0, 1, 5556725, 193926796, 119843417, 5289611, 50985, 388, 8, 1, 1

Offset: 0

Andrew Howroyd, Jan 10 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 binary matrices with k distinct columns and any number of nonzero rows with n ones in every column and rows in nonincreasing lexicographic order.

			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   9     70       794         12055            233238 ...
  3 | 1 1 3  29    666     28344       1935054         193926796 ...
  4 | 1 1 4  68   3642    469368     119843417       53059346010 ...
  5 | 1 1 5 134  14951   5289611    4681749424     8639480647842 ...
  6 | 1 1 6 237  50985  46241343  134332244907   989821806791367 ...
  7 | 1 1 7 388 151901 333750928 3032595328876 85801167516707734 ...
     ...
The A(2,2) = 2 matrices are:
   [1 1]   [1 0]
   [1 0]   [1 0]
   [0 1]   [0 1]
           [0 1]
The corresponding set multipartitions are:
    {{1,2}, {1}, {2}},
    {{1}, {1}, {2}, {2}}.

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

Rows n=1..3 are A000012, A014500, A331389.
Columns k=0..3 are A000012, A000012, A001477, A331390.
Cf. A188392, A188445, A330942, A331039, A331160, A331161, A331391.

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]]++); binomial(WeighT(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=1..k} Stirling1(k, j)*A188392(n, j) for n, k >= 1.

A331391(n) = Sum_{d|n} A(n/d, d).

A331160 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of distinct nonzero rows with column sums n and rows in decreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 2, 1, 0, 1, 27, 15, 2, 1, 0, 1, 266, 317, 44, 3, 1, 0, 1, 3599, 12586, 2763, 120, 4, 1, 0, 1, 62941, 803764, 390399, 21006, 319, 5, 1, 0, 1, 1372117, 75603729, 103678954, 10074052, 147296, 804, 6, 1

Offset: 0

Andrew Howroyd, Jan 10 2020

The condition that the rows be in decreasing order is equivalent to considering nonequivalent matrices with distinct rows 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 1   4     27        266           3599            62941 ...
  3 | 1 2  15    317      12586         803764         75603729 ...
  4 | 1 2  44   2763     390399      103678954      46278915417 ...
  5 | 1 3 120  21006   10074052    10679934500   21806685647346 ...
  6 | 1 4 319 147296  232165926   956594630508 8717423133548684 ...
  7 | 1 5 804 967829 4903530137 76812482919237 ...
      ...
The A(2,2) = 4 matrices are:
   [2 1]   [2 0]   [1 2]   [1 1]
   [0 1]   [0 2]   [1 0]   [1 0]
                           [0 1]

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

Rows n=1..3 are A000012, A331316, A331344
Columns k=0..2 are A000012, A000009, A331317.
Cf. A219585, A331039, A331126, A331161, A331315, A331318.

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+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, k<=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)}

A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A219585(n, j).

A331318(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).

A059530 Triangle T(n,k) of k-block T_0-tricoverings of an n-set, n >= 3, k = 0..2*n.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 39, 89, 43, 3, 0, 0, 0, 0, 0, 252, 2192, 4090, 2435, 445, 12, 0, 0, 0, 0, 0, 1260, 37080, 179890, 289170, 188540, 50645, 4710, 70, 0, 0, 0, 0, 0, 5040, 536760, 6052730, 20660055, 29432319, 19826737, 6481160, 964495, 52430

Offset: 3

Vladeta Jovovic, Feb 22 2001

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering. A covering of a set is a T_0-covering if for every two distinct elements of the set there exists a block of the covering containing one but not the other element.

			Triangle begins:
  [0, 0, 0, 0, 1, 3, 1],
  [0, 0, 0, 0, 1, 39, 89, 43, 3],
  [0, 0, 0, 0, 0, 252, 2192, 4090, 2435, 445, 12],
  [0, 0, 0, 0, 0, 1260, 37080, 179890, 289170, 188540, 50645, 4710, 70],
   ...
There are 5 = 1 + 3 + 1 T_0-tricoverings of a 3-set and 175 = 1 + 39 + 89 + 43 + 3 T_0-tricoverings of a 4-set, cf. A060070.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Andrew Howroyd, Table of n, a(n) for n = 3..1674
Vladeta Jovovic, T_0-tricoverings of a 4-set

Column sums are A060069.
Row sums are A060070.
Cf. A059443, A060052, A060092, A060487, A060492, A331039.

\\ gets k-th column as vector
C(k)=if(k<4, [], Vecrev(serlaplace(polcoef(exp(-x + x^2/2 + x^3*y/3 + O(x*x^k))*sum(i=0, 2*k, (1+y)^binomial(i, 3)*exp(-x^2*(1+y)^i/2 + O(x*x^k))*x^i/i!), k))/y)) \\ Andrew Howroyd, Jan 30 2020

T(n)={my(m=2*n, y='y + O('y^(n+1))); my(g=exp(-x + x^2/2 + x^3*y/3 + O(x*x^m))*sum(k=0, m, (1+y)^binomial(k, 3)*exp(-x^2*(1+y)^k/2 + O(x*x^m))*x^k/k!)); Mat([Col(serlaplace(p), -n) | p<-Vec(g)[2..m+1]]);}
{ my(A=T(8)); for(n=3, matsize(A)[1], print(concat([0], A[n, 1..2*n]))) } \\ Andrew Howroyd, Jan 30 2020

E.g.f. for k-block T_0-tricoverings of an n-set is exp(-x+1/2*x^2+1/3*x^3*y)*Sum_{i=0..inf}(1+y)^binomial(i, 3)*exp(-1/2*x^2*(1+y)^i)*x^i/i!.

T(n,k) = 0 for n > binomial(k, 3). - Andrew Howroyd, Jan 30 2020

cp's OEIS Frontend

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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A060053 Number of r-bicoverings (or restricted proper 2-covers) of an n-set.

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A188445 T(n,k) is the number of (n*k) X k binary arrays with nonzero rows in decreasing order and n ones in every column.

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A060070 Number of T_0-tricoverings of an n-set.

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A060052 Triangle read by rows: T(n,k) gives number of r-bicoverings of an n-set with k blocks, n >= 2, k = 3..n+floor(n/2).

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A331569 Array read by antidiagonals: A(n,k) is the number of binary matrices with k distinct columns and any number of distinct nonzero rows with n ones in every column and columns in decreasing lexicographic order.

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A331126 Array read by antidiagonals: A(n,k) is the number of T_0 n-regular set multipartitions (multisets of sets) on a k-set.

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