A048291 -id:A048291 - cp's OEIS frontend

A183109 Triangle read by rows: T(n,m) = number of n X m binary matrices with no zero rows or columns (n >= 1, 1 <= m <= n).

1, 1, 7, 1, 25, 265, 1, 79, 2161, 41503, 1, 241, 16081, 693601, 24997921, 1, 727, 115465, 10924399, 831719761, 57366997447, 1, 2185, 816985, 167578321, 26666530801, 3776451407065, 505874809287625

Offset: 1

Steffen Eger, Feb 01 2011

T(n,m) = T(m,n) is also the number of complete alignments between two strings of sizes m and n, respectively; i.e. the number of complete matchings in a bipartite graph
From Manfred Boergens, Jul 25 2021: (Start)
The matrices in the definition are a superset of the matrices in the comment to A019538 by Manfred Boergens.
T(n,m) is the number of coverings of [n] by tuples (A_1,...,A_m) in P([n])^m with nonempty A_j, with P(.) denoting the power set (corrected for clarity by Manfred Boergens, May 26 2024). For the disjoint case see A019538.
For tuples with "nonempty" dropped see A092477 and A329943 (amendment by Manfred Boergens, Jun 24 2024). (End)

			Triangle begins:
  1;
  1,    7;
  1,   25,    265;
  1,   79,   2161,     41503;
  1,  241,  16081,    693601,    24997921;
  1,  727, 115465,  10924399,   831719761,   57366997447;
  1, 2185, 816985, 167578321, 26666530801, 3776451407065, 505874809287625;
  ...

Indranil Ghosh, Rows 1..50, flattened
Ch. A. Charalambides, A problem of arrangements on chessboards and generalizations, Discrete Mathematics 27.2 (1979): 179-186. (Generalizations.)
D. E. Knuth, Problem 11243, Am. Math. Montly 113 (8) (2006) page 759.
John Riordan and Paul R. Stein, Arrangements on chessboards, Journal of Combinatorial Theory, Series A 12.1 (1972): 72-80. See Table page 78.

Cf. A218695 (same sequence with restriction m<=n dropped).
Cf. A058482 (this gives the general formula, but values only for m=3).
Main diagonal gives A048291.
Column 2 is A058481.
Cf. A019538, A092477, A329943.

A183109 := proc(n,m)
    add((-1)^j*binomial(m,j)*(2^(m-j)-1)^n,j=0..m) ;
end proc:
seq(seq(A183109(n,m),m=1..n),n=1..10) ; # R. J. Mathar, Dec 03 2015

Flatten[Table[Sum[ (-1)^j*Binomial[m, j]*(2^(m - j) - 1)^n, {j, 0, m}], {n, 1, 7}, {m, 1, n}]] (* Indranil Ghosh, Mar 14 2017 *)

tabl(nn) = {for(n=1, nn, for(m = 1, n, print1(sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n),", ");); print(););};
tabl(8); \\ Indranil Ghosh, Mar 14 2017

import math
f = math.factorial
def C(n,r): return f(n)//f(r)//f(n - r)
def T(n,m):
    return sum((-1)**j*C(m,j)*(2**(m - j) - 1)**n for j in range (m+1))
i=1
for n in range(1,21):
    for m in range(1, n+1):
        print(str(i)+" "+str(T(n, m)))
        i+=1 # Indranil Ghosh, Mar 14 2017

T(n,m) = Sum_{j=0..m}(-1)^j*C(m, j)*(2^(m-j)-1)^n.
Recursion: T(m,n) = Sum_{k=1..m} T(k,n-1)*C(m,k)*2^k - T(m,n-1).
From Robert FERREOL, Mar 14 2017: (Start)
T(n,m) = Sum_{i = 0 .. n,j = 0 ..m}(-1)^(n+m+i+j)*C(n,i)*C(m,j)*2^(i*j).
Inverse formula of: 2^(n*m) = Sum_{i = 0 .. n , j = 0 ..m} C(n,i)*C(m,j)*T(i,j). (End)
A019538(j) <= a(j). - Manfred Boergens, Jul 25 2021

A054780 Number of n-covers of a labeled n-set.

Original entry on oeis.org

1, 1, 3, 32, 1225, 155106, 63602770, 85538516963, 386246934638991, 6001601072676524540, 327951891446717800997416, 64149416776011080449232990868, 45546527789182522411309599498741023, 118653450898277491435912500458608964207578

Offset: 0

Vladeta Jovovic, May 21 2000

Also, number of n X n rational {0,1}-matrices with no zero rows or columns and with all rows distinct, up to permutation of rows.

			From _Gus Wiseman_, Dec 19 2023: (Start)
Number of ways to choose n nonempty sets with union {1..n}. For example, the a(3) = 32 covers are:
  {1}{2}{3}  {1}{2}{13}  {1}{2}{123}  {1}{12}{123}  {12}{13}{123}
             {1}{2}{23}  {1}{3}{123}  {1}{13}{123}  {12}{23}{123}
             {1}{3}{12}  {1}{12}{13}  {1}{23}{123}  {13}{23}{123}
             {1}{3}{23}  {1}{12}{23}  {2}{12}{123}
             {2}{3}{12}  {1}{13}{23}  {2}{13}{123}
             {2}{3}{13}  {2}{3}{123}  {2}{23}{123}
                         {2}{12}{13}  {3}{12}{123}
                         {2}{12}{23}  {3}{13}{123}
                         {2}{13}{23}  {3}{23}{123}
                         {3}{12}{13}  {12}{13}{23}
                         {3}{12}{23}
                         {3}{13}{23}
(End)

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

Main diagonal of A055154.
Cf. A048291, A088310, A181230, A259763.
Covers with any number of edges are counted by A003465, unlabeled A055621.
Connected graphs of this type are counted by A057500, unlabeled A001429.
This is the covering case of A136556.
The case of graphs is A367863, covering case of A116508, unlabeled A006649.
Binomial transform is A367916.
These set-systems have ranks A367917.
The unlabeled version is A368186.
A006129 counts covering graphs, connected A001187, unlabeled A002494.
A046165 counts minimal covers, ranks A309326.
Cf. A006126, A058891, A059201, A133686, A323816, A323817, A323818, A326754, A367862, A367901.

Join[{1}, Table[Sum[StirlingS1[n+1, k+1]*(2^k - 1)^n, {k, 0, n}]/n!, {n, 1, 15}]] (* Vaclav Kotesovec, Jun 04 2022 *)
Table[Length[Select[Subsets[Rest[Subsets[Range[n]]],{n}],Union@@#==Range[n]&]],{n,0,4}] (* Gus Wiseman, Dec 19 2023 *)

a(n) = sum(k=0, n, (-1)^k*binomial(n, k)*binomial(2^(n-k)-1, n)) \\ Andrew Howroyd, Jan 20 2024

a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*binomial(2^(n-k)-1, n).
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n+1, k+1)*(2^k-1)^n.
G.f.: Sum_{n>=0} log(1+(2^n-1)*x)^n/((1+(2^n-1)*x)*n!). - Paul D. Hanna and Vladeta Jovovic, Jan 16 2008
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jun 04 2022
Inverse binomial transform of A367916. - Gus Wiseman, Dec 19 2023

A057150 Triangle read by rows: T(n,k) = number of k X k binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 5, 2, 1, 0, 0, 4, 11, 2, 1, 0, 0, 3, 21, 14, 2, 1, 0, 0, 1, 34, 49, 15, 2, 1, 0, 0, 1, 33, 131, 69, 15, 2, 1, 0, 0, 0, 33, 248, 288, 79, 15, 2, 1, 0, 0, 0, 19, 410, 840, 420, 82, 15, 2, 1, 0, 0, 0, 14, 531, 2144, 1744, 497, 83, 15, 2, 1

Offset: 1

Vladeta Jovovic, Aug 14 2000

Also the number of non-isomorphic set multipartitions (multisets of sets) of weight n with k parts and k vertices. - Gus Wiseman, Nov 14 2018

			[1], [0,1], [0,1,1], [0,1,2,1], [0,0,5,2,1], [0,0,4,11,2,1], ...;
There are 8 square binary matrices with 5 ones, with no zero rows or columns, up to row and column permutation: 5 of size 3 X 3:
[0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1]
[0 0 1] [0 1 0] [0 1 1] [0 1 1] [1 1 0]
[1 1 1] [1 1 1] [1 0 1] [1 1 0] [1 1 0]
2 of size 4 X 4:
[0 0 0 1] [0 0 0 1]
[0 0 0 1] [0 0 1 0]
[0 0 1 0] [0 1 0 0]
[1 1 0 0] [1 0 0 1]
and 1 of size 5 X 5:
[0 0 0 0 1]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
[1 0 0 0 0].
From _Gus Wiseman_, Nov 14 2018: (Start)
Triangle begins:
   1
   0   1
   0   1   1
   0   1   2   1
   0   0   5   2   1
   0   0   4  11   2   1
   0   0   3  21  14   2   1
   0   0   1  34  49  15   2   1
   0   0   1  33 131  69  15   2   1
   0   0   0  33 248 288  79  15   2   1
Non-isomorphic representatives of the multiset partitions counted in row 6 {0,0,4,11,2,1} are:
  {{12}{13}{23}}  {{1}{1}{1}{234}}  {{1}{2}{3}{3}{45}}  {{1}{2}{3}{4}{5}{6}}
  {{1}{23}{123}}  {{1}{1}{24}{34}}  {{1}{2}{3}{5}{45}}
  {{13}{23}{23}}  {{1}{1}{4}{234}}
  {{3}{23}{123}}  {{1}{2}{34}{34}}
                  {{1}{3}{24}{34}}
                  {{1}{3}{4}{234}}
                  {{1}{4}{24}{34}}
                  {{1}{4}{4}{234}}
                  {{2}{4}{12}{34}}
                  {{3}{4}{12}{34}}
                  {{4}{4}{12}{34}}
(End)

Row sums give A057151.
Cf. A049311, A056037, A056079, A056080, A057149, A057151, A057152.
Cf. A007716, A048291, A054976, A101370, A104601, A104602, A120732, A120733, A135588, A319616, A321609, A321615.

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[Product[Product[(1 + O[x]^(k + 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_] := 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!)];
T[n_, k_] := M[k, k, n] - 2*M[k, k - 1, n] + M[k - 1, k - 1, n];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 10 2019, after Andrew Howroyd *)

\\ See A321609 for M.
T(n,k) = M(k,k,n) - 2*M(k,k-1,n) + M(k-1,k-1,n); \\ Andrew Howroyd, Nov 14 2018

Duplicate seventh row removed by Gus Wiseman, Nov 14 2018

A262307 Array read by antidiagonals: T(m,n) = number of m X n binary matrices with all 1's connected and no zero rows or columns.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 19, 19, 1, 1, 65, 205, 65, 1, 1, 211, 1795, 1795, 211, 1, 1, 665, 14221, 36317, 14221, 665, 1, 1, 2059, 106819, 636331, 636331, 106819, 2059, 1, 1, 6305, 778765, 10365005, 23679901, 10365005, 778765, 6305, 1

Offset: 1

N. J. A. Sloane, Oct 04 2015

Two 1's are connected if they are in the same row or column. The requirement is for them to form a single connected set.
The number of m X n binary matrices with no zero rows or columns is given by A183109(m, n). If there are multiple components (not connected) then they cannot share either rows or columns. For i < n and j < m there are T(i,j) ways of creating an i X j component that occupies the first row. Its remaining i-1 rows may be on any of the remaining m-1 rows with its j columns on any of the n columns. The m-i rows and n-j columns not used by this component can be any matrix with no zero rows or columns.
T(m,n) is also the number of bipartite connected labeled graphs with parts of size m and n. (See A005333, A227322.)
This is the array a(m,n) in Kreweras (1969). Kreweras describes this as a symmetric triangle read by rows, giving numbers of connected relations.
The companion array b(m,n) (and the first few of its diagonals) in Kreweras (1969) should also be added to the OEIS if they are not already present.

			Table starts:
==========================================================================
m\n| 1    2      3         4           5             6               7
---|----------------------------------------------------------------------
1  | 1    1      1         1           1             1               1 ...
2  | 1    5     19        65         211           665            2059 ...
3  | 1   19    205      1795       14221        106819          778765 ...
4  | 1   65   1795     36317      636331      10365005       162470155 ...
5  | 1  211  14221    636331    23679901     805351531     26175881341 ...
6  | 1  665 106819  10365005   805351531   56294206205   3735873535339 ...
7  | 1 2059 778765 162470155 26175881341 3735873535339 502757743028605 ...
...
As a triangle, this begins:
  1;
  1,    1;
  1,    5,      1;
  1,   19,     19,      1;
  1,   65,    205,     65,      1;
  1,  211,   1795,   1795,    211,      1;
  1,  665,  14221,  36317,  14221,    665,    1;
  1, 2059, 106819, 636331, 636331, 106819, 2059, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..780
G. Kreweras, Inversion des polynômes de Bell bidimensionnels et application au dénombrement des relations binaires connexes, C. R. Acad. Sci. Paris Ser. A-B 268 1969 A577-A579.

Essentially same table as triangle A227322 (where the antidiagonals only go halfway).
Main diagonal is A005333.
Initial diagonals give A001047, A002501, A002502.
Cf. A226658, A123260, A286189, A183109, A048291, A287274.

A183109[n_, m_] := Sum[(-1)^j*Binomial[m, j]*(2^(m-j) - 1)^n, {j, 0, m}];
T[m_, n_] := A183109[m, n] - Sum[T[i, j]*A183109[m - i, n - j] Binomial[m - 1, i - 1]*Binomial[n, j], {i, 1, m - 1}, {j, 1, n - 1}];
Table[T[m - n + 1, n], {m, 1, 9}, {n, 1, m}] // Flatten (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

G(N)={my(S=matrix(N,N), T=matrix(N,N));
for(m=1,N,for(n=1,N,
S[m,n]=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
T[m,n]=S[m,n]-sum(i=1, m-1, sum(j=1, n-1, T[i,j]*S[m-i,n-j]*binomial(m-1,i-1)*binomial(n,j)));
));T}
G(7) \\ Andrew Howroyd, May 22 2017

T(m,n) = A183109(m,n) - Sum_{i=1..m-1} Sum_{j=1..n-1} T(i,j)*A183109(m-i, n-j)*binomial(m-1,i-1)*binomial(n,j). - Andrew Howroyd, May 22 2017

Revised by N. J. A. Sloane, May 26 2017, to incorporate material from Andrew Howroyd's May 22 2017 submission (formerly A287297), which was essentially identical to this, although giving an alternative description and more information.

A104601 Triangle T(r,n) read by rows: number of n X n (0,1)-matrices with exactly r entries equal to 1 and no zero row or columns.

Original entry on oeis.org

1, 0, 2, 0, 4, 6, 0, 1, 45, 24, 0, 0, 90, 432, 120, 0, 0, 78, 2248, 4200, 720, 0, 0, 36, 5776, 43000, 43200, 5040, 0, 0, 9, 9066, 222925, 755100, 476280, 40320, 0, 0, 1, 9696, 727375, 6700500, 13003620, 5644800, 362880, 0, 0, 0, 7480, 1674840

Offset: 1

Ralf Stephan, Mar 27 2005

			1
0,2
0,4,6
0,1,45,24
0,0,90,432,120
0,0,78,2248,4200,720
0,0,36,5776,43000,43200,5040
0,0,9,9066,222925,755100,476280,40320
0,0,1,9696,727375,6700500,13003620,5644800,362880
0,0,0,7480,1674840,37638036,179494350,226262400,71850240,3628800

M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.

Right-edge diagonals include A000142, A055602, A055603. Row sums are in A104602.
Column sums are in A048291. The triangle read by columns = A055599.
Cf. A049311, A054976, A057150, A057151, A101370, A120732, A120733, A138178, A316983, A319616.

t[r_, n_] := Sum[ Sum[ (-1)^(2n - d - k/d)*Binomial[n, d]*Binomial[n, k/d]*Binomial[k, r], {d, Divisors[k]}], {k, r, n^2}]; Flatten[ Table[t[r, n], {r, 1, 10}, {n, 1, r}]] (* Jean-François Alcover, Jun 27 2012, from formula *)
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],Union[First/@#]==Union[Last/@#]==Range[k]&]],{n,6},{k,n}] (* Gus Wiseman, Nov 14 2018 *)

T(r, n) = Sum{l>=r, Sum{d|l, (-1)^(2n-d-l/d)*C(n, d)*C(n, l/d)*C(l, r) }}.

E.g.f.: Sum(((1+x)^n-1)^n*exp((1-(1+x)^n)*y)*y^n/n!,n=0..infinity). - Vladeta Jovovic, Feb 24 2008

A086193 Number of n X n matrices with entries in {0,1} with no zero row, no zero column and with zero main diagonal.

Original entry on oeis.org

1, 0, 1, 18, 1699, 592260, 754179301, 3562635108438, 63770601591579079, 4405870283636411477640, 1190873924687350003735546441, 1270602397076493907445608866890778, 5381240610642043789096251476993474339179

Offset: 0

W. Edwin Clark, Aug 25 2003

nonn

Also the number of simple labeled digraphs on n nodes for which every vertex has indegree at least one and outdegree at least one.

Also the number of edge covers on the n-crown graph. - Eric W. Weisstein, May 19 2017

Andrew Howroyd, Table of n, a(n) for n = 0..50
R. W. Robinson. Counting digraphs with restrictions on the strong components, (1996). W(n).
R. W. Robinson, Counting digraphs with restrictions on the strong components, 1996 [Local copy, with permission]
Martin Svatoš, Peter Jung, Jan Tóth, Yuyi Wang, and Ondřej Kuželka, On Discovering Interesting Combinatorial Integer Sequences, arXiv:2302.04606 [cs.LO], 2023, p. 17.
Eric Weisstein's World of Mathematics, Crown Graph
Eric Weisstein's World of Mathematics, Edge Cover

Cf. A048291.

Table[ it = (Partition[ #1, n ] &) /@ IntegerDigits[ Range[ 0, -1 + 2^n^2 ], 2, n^2 ]; Count[ it, (q_)?MatrixQ /; Tr[ q ] === 0 && (Times @@ (Plus @@@ q)) > 0 && (Times @@ (Plus @@@ Transpose[ q ]) > 0) ], {n, 1, 4} ] (* Wouter Meeussen, Aug 25 2003 *)
Table[Sum[(-1)^(n-r)*Binomial[n, r]*(2^(r-1)-1)^r*(2^r-1)^(n-r), {r,0,n}],{n,1,15}] (* Vaclav Kotesovec, May 04 2015 after Vladeta Jovovic *)

a(n)={sum(r=0, n, (-1)^(n-r)*binomial(n, r)*(2^(r-1)-1)^r*(2^r-1)^(n-r))} \\ Andrew Howroyd, Sep 09 2018

a(n) = Sum_{r=0..n} (-1)^(n-r)*binomial(n, r)*(2^(r-1)-1)^r*(2^r-1)^(n-r). - Vladeta Jovovic, Aug 27 2003
a(n) = sum( f(n, r), r=0..n ) where f(n, r) = binomial(n, r) (-1)^r (1-2^(-n+r+1))^(n-r) (1-2^(-n+r))^r 2^((n-r)(n-1)). - Brendan McKay, Aug 27 2003
E.g.f.: Sum_{k>=0} (2^(n-1)-1)^n*exp((1-2^n)*x)*x^n/n!. - Vladeta Jovovic, Feb 23 2008
a(n) ~ 2^(n*(n-1)). - Vaclav Kotesovec, May 04 2015

More terms from Brendan McKay, Aug 27 2003

a(0)=1 prepended by Andrew Howroyd, Sep 09 2018

A218695 Square array A(h,k) = (2^h-1)A(h,k-1) + Sum_{i=1..h-1} binomial(h,h-i)2^i*A(i,k-1), with A(1,k) = A(h,1) = 1; read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 25, 25, 1, 1, 79, 265, 79, 1, 1, 241, 2161, 2161, 241, 1, 1, 727, 16081, 41503, 16081, 727, 1, 1, 2185, 115465, 693601, 693601, 115465, 2185, 1, 1, 6559, 816985, 10924399, 24997921, 10924399, 816985, 6559, 1

Offset: 1

M. F. Hasler, Nov 04 2012

This symmetric table is defined in the Kreweras papers, used also in A223911. Its upper or lower triangular part equals A183109, which might provide a simpler formula.
Number of h X k binary matrices with no zero rows or columns. - Andrew Howroyd, Mar 29 2023
A(h,k) is the number of coverings of [h] by tuples (A_1,...,A_k) in P([h])^k with nonempty A_j, with P(.) denoting the power set. For the disjoint case see A019538. For tuples with "nonempty" omitted see A092477 and A329943 (amendment by Manfred Boergens, Jun 24 2024). - Manfred Boergens, May 26 2024

			Array A(h,k) begins:
=====================================================
h\k | 1   2      3        4         5           6 ...
----+------------------------------------------------
  1 | 1   1      1        1         1           1 ...
  2 | 1   7     25       79       241         727 ...
  3 | 1  25    265     2161     16081      115465 ...
  4 | 1  79   2161    41503    693601    10924399 ...
  5 | 1 241  16081   693601  24997921   831719761 ...
  6 | 1 727 115465 10924399 831719761 57366997447 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
G. Kreweras, Dénombrements systématiques de relations binaires externes, Math. Sci. Humaines 26 (1969) 5-15.
G. Kreweras, Dénombrement des ordres étagés, Discrete Math., 53 (1985), 147-149.

Columns 1..3 are A000012, A058481, A058482.
Main diagonal is A048291.
Cf. A019538, A056152 (unlabeled case), A052332, A092477, A183109, A223911, A329943.

c(h,k)={(h<2 || k<2) & return(1); sum(i=1,h-1,binomial(h,h-i)*2^i*c(i,k-1))+(2^h-1)*c(h,k-1)}
/* For better performance when h and k are large, insert the following memoization code before "sum(...)": cM=='cM & cM=matrix(h,k); my(s=matsize(cM));
s[1] >= h & s[2] >= k & cM[h,k] & return(cM[h,k]);
s[1]

A(m, n) = sum(k=0, m, (-1)^(m-k) * binomial(m, k) * (2^k-1)^n ) \\ Andrew Howroyd, Mar 29 2023

From Andrew Howroyd, Mar 29 2023: (Start)
A(h, k) = Sum_{i=0..h} (-1)^(h-i) * binomial(h, i) * (2^i-1)^k.
A052332(n) = Sum_{i=1..n-1} binomial(n,i)*A(i, n-i) for n > 0. (End)

A287065 Number of dominating sets on the n X n rook graph.

Original entry on oeis.org

1, 11, 421, 59747, 32260381, 67680006971, 559876911043381, 18412604442711949187, 2416403019417984915336061, 1267413006543912045144741284411, 2658304092145691708492995820522716981, 22300364428188338185156192161829091442585827

Offset: 1

Eric W. Weisstein, May 19 2017

nonn

Number of {0,1} n X n matrices with no zero rows or no zero columns. - Geoffrey Critzer, Jan 15 2024

Andrew Howroyd, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Dominating Set
Eric Weisstein's World of Mathematics, Rook Graph

Main diagonal of A287274.
Row sums of A368831.
Cf. A183109, A002720, A048291, A173403.

Table[(2^n - 1)^n + Sum[Binomial[n, i] Sum[(-1)^j (-1 + 2^(n - j))^i Binomial[n, j], {j, 0, n}], {i, n - 1}], {n, 20}] (* Eric W. Weisstein, May 27 2017 *)

b(m,n)=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
a(n)=(2^n-1)^n + sum(i=1,n-1,b(n,i)*binomial(n,i)); \\ Andrew Howroyd, May 22 2017

a(n) = (2^n-1)^n + Sum_{i=1..n-1} binomial(n,i) * A183109(n,i). - Andrew Howroyd, May 22 2017

a(6)-a(12) from Andrew Howroyd, May 22 2017

A055599 Triangle T(n,k) read by rows, giving the number of n X n binary matrices with no zero rows or columns and with k=0..n^2 ones.

Original entry on oeis.org

0, 1, 0, 0, 2, 4, 1, 0, 0, 0, 6, 45, 90, 78, 36, 9, 1, 0, 0, 0, 0, 24, 432, 2248, 5776, 9066, 9696, 7480, 4272, 1812, 560, 120, 16, 1, 0, 0, 0, 0, 0, 120, 4200, 43000, 222925, 727375, 1674840, 2913100, 3995100, 4441200, 4073100, 3114140, 1994550

Offset: 1

Vladeta Jovovic, Jun 01 2000

Rows also give the coefficients of the edge cover polynomials of the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017

			For m=n=3 we get T(3,k)=C(9,k)-6*C(6,k)+9*C(4,k)+6*C(3,k)-18*C(2,k)+9*C(1,k)-C(0,k) giving the batch [0,0,0,6,45,90,78,36,9,1].
Triangle begins:
0, 1,
0, 0, 2, 4, 1,
0, 0, 0, 6, 45, 90, 78, 36, 9, 1,
0, 0, 0, 0, 24, 432, 2248, 5776, 9066, 9696, 7480, 4272, 1812, 560, 120, 16, 1,
...

H. Cheballah, S. Giraudo, and R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.
Stephan Mertens, Domination Polynomial of the Rook Graph, JIS 27 (2024) 24.3.7; arXiv:2401.00716 [math.CO], 2024.
Roberto Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Edge Cover Polynomial

Cf. A048291 (row sums).

row[n_] := Sum[(-1)^(n-k) Binomial[n, k] ((1+x)^k - 1)^n, {k, 0, n}] + O[x]^(n^2+1) // CoefficientList[#, x]&;
Table[row[n], {n, 1, 5}] // Flatten (* Jean-François Alcover, Nov 06 2018 *)

Number of m X n binary matrices with no zero rows or columns and with k=0..m*n ones is Sum_{i=0..n} (-1)^i*C(n, i)*a(m, n-i, k) where a(m, n, k)=Sum_{i=0..m} (-1)^i*C(m, i)*C((m-i)*n, k).
G.f. for n-th row: Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*((1+x)^k-1)^n. - Vladeta Jovovic, Apr 04 2003
E.g.f.: Sum(((1+y)^n-1)^n*exp((1-(1+y)^n)*x)*x^n/n!,n=0..infinity). - Vladeta Jovovic, Feb 24 2008

A055084 Number of 6 X n binary matrices with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 15, 180, 2001, 20755, 200082, 1781941, 14637962, 111011667, 779695050, 5093379110, 31092553357, 178203364143, 963217652830, 4930357535218, 23989343505296, 111335037107474, 494383391324356, 2106346854756098

Offset: 1

Vladeta Jovovic, Jun 13 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500

Column k=6 of A056152.

Cf. A024206, A055609, A054976, A048291.

Vec((G(6, x) - G(5, x))*(1 - x) + O(x^30)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

cp's OEIS Frontend

A183109 Triangle read by rows: T(n,m) = number of n X m binary matrices with no zero rows or columns (n >= 1, 1 <= m <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A054780 Number of n-covers of a labeled n-set.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A057150 Triangle read by rows: T(n,k) = number of k X k binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A262307 Array read by antidiagonals: T(m,n) = number of m X n binary matrices with all 1's connected and no zero rows or columns.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A104601 Triangle T(r,n) read by rows: number of n X n (0,1)-matrices with exactly r entries equal to 1 and no zero row or columns.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A086193 Number of n X n matrices with entries in {0,1} with no zero row, no zero column and with zero main diagonal.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A218695 Square array A(h,k) = (2^h-1)*A(h,k-1) + Sum_{i=1..h-1} binomial(h,h-i)*2^i*A(i,k-1), with A(1,k) = A(h,1) = 1; read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

A218695 Square array A(h,k) = (2^h-1)A(h,k-1) + Sum_{i=1..h-1} binomial(h,h-i)2^i*A(i,k-1), with A(1,k) = A(h,1) = 1; read by antidiagonals.