A055603 -id:A055603 - cp's OEIS frontend

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.

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

A055602 Number of n X n binary matrices with no 0 rows or columns and with n+1 1's.

Original entry on oeis.org

0, 4, 45, 432, 4200, 43200, 476280, 5644800, 71850240, 979776000, 14270256000, 221298739200, 3642807168000, 63465795993600, 1167099373440000, 22596613079040000, 459548157100032000, 9795631769763840000

Offset: 1

Vladeta Jovovic, Jun 01 2000

nonn

Robert Israel, Table of n, a(n) for n = 1..445

A diagonal of triangle A104601.

Cf. A055603.

f:= n -> n*(n-1)*(n+2)*n!/4:
map(f, [$1..30]); # Robert Israel, May 04 2021

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*binomial(n, i)*a(m, n-i, k) where a(m, n, k)=Sum_{i=0..m} (-1)^i*binomial(m, i)*binomial((m-i)*n, k).
a(n) = n*(n-1)*(n+2)*n!/4. - Vladeta Jovovic, Mar 25 2006
From Robert Israel, May 04 2021: (Start)
E.g.f.: x^2*(4-x)/(2*(1-x)^2).
D-finite with recurrence 4*(n-2)*a(n)-n*(4*n+3)*a(n-1)-(n-1)^2*a(n-2)=0.
(End)

More terms from David Wasserman, Apr 28 2002

A084485 Number of 3 X n 0-1 matrices which have n+2 1's and have no zero rows or zero columns.

Original entry on oeis.org

1, 12, 90, 522, 2595, 11673, 49014, 195828, 753813, 2819475, 10308144, 36998118, 130786695, 456452493, 1575799290, 5389290792, 18281487081, 61569776727, 206040460212, 685584843450, 2269566343611, 7478425876977, 24538396875870, 80206515476892, 261239771497725

Offset: 1

W. Edwin Clark, May 27 2003

This is the number of spanning subgraphs of the complete bipartite graph K(3,n) with n + 2 edges and no isolated vertices. If the subgraphs are also connected then they are spanning trees. The number of spanning trees in K(m,n) is known. See A001787.

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5
Index entries for linear recurrences with constant coefficients, signature (15,-93,305,-558,540,-216)

Cf. A001787.

Cf. A084486, A055602, A055603.

with(LinearAlgebra): num1s:= (M, m, n)->add(ListTools[Flatten](convert(M, listlist))[j], j=1..m*n): binrows:= n->[seq(convert(i+2^n, base, 2)[1..n], i=1..2^n-1)]: a:= proc(n) local A, L, i, j, k, S, M: S := 0: L := binrows(n): for i from 1 to 2^n-1 do for j from 1 to 2^n-1 do for k from 1 to 2^n-1 do A := Matrix([L[i], L[j], L[k]]); if num1s(A, 3, n)=n+2 and (not has(Matrix([1, 1, 1]).A, 0)) then S := S+1; end if; od; od; od; S; end proc: seq (a(n), n=1..5);

a[n_] := n*(4*(3*n - 1)*3^n - 9*(n - 1)*2^n)/24;
Array[a, 25] (* Jean-François Alcover, Nov 10 2017, after Vladeta Jovovic *)

a(n) = n*(4*(3*n-1)*3^n-9*(n-1)*2^n)/24. - Vladeta Jovovic, May 28 2003

G.f.: x*(1-3*x+3*x^2-17*x^3+33*x^4)/((3*x-1)^3*(2*x-1)^3). - Alois P. Heinz, Sep 24 2012

Comment corrected by W. Edwin Clark, Sep 24 2012

A084486 Number of 4 X n 0-1 matrices which have n+3 1's and have no zero rows or zero columns.

Original entry on oeis.org

1, 32, 522, 5776, 50600, 380424, 2570932, 16073600, 94748400, 533515240, 2896652396, 15268777440, 78544641448, 395875164104, 1960998472260, 9570684204544, 46112171619296, 219682468794600, 1036237335593500

Offset: 1

W. Edwin Clark, May 27 2003

This is the number of spanning subgraphs of the complete bipartite graph K(4,n) which have n+3 edges and no isolated vertices. If the subgraphs are also connected then they are spanning trees. The number of spanning trees in K(m,n) is known. See A001787.

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5

Cf. A001787.

Cf. A084485, A055602, A055603.

with(LinearAlgebra): num1s := (M,m,n)->add(ListTools[Flatten](convert(M,listlist))[j],j=1..m*n): binrows := n->[seq(convert(i+2^n,base,2)[1..n],i=1..2^n-1)]; a := proc(n) local A,L,i,j,k,el,S,M: S := 0: L := binrows(n): for i from 1 to 2^n-1 do for j from 1 to 2^n-1 do for k from 1 to 2^n-1 do for el from 1 to 2^n-1 do A := Matrix([L[i],L[j],L[k],L[el]]); if num1s(A,4,n)=n+3 and (not has(Matrix([1,1,1,1]).A,0)) then S := S+1; end if; od; od; od; od; S; end proc: seq (a(n), n=1..2);

a[n_] := n/48*((27*4^n - 32*3^n + 6*2^n)*n^2 + (-9*4^n + 32*3^n - 18*2^n)*n + (-6*4^n + 12*2^n));
Array[a, 20] (* Jean-François Alcover, Nov 10 2017, after Vladeta Jovovic *)

n/48*((27*4^n-32*3^n+6*2^n)*n^2+(-9*4^n+32*3^n-18*2^n)*n+(-6*4^n+12*2^n)). - Vladeta Jovovic, May 28 2003

G.f.: x * (1 -4*x -40*x^2 +44*x^3 +2885*x^4 -19624*x^5 +59014*x^6 -97728*x^7 +98064*x^8 -67200*x^9 +28800*x^10) / ((3*x-1)^4*(2*x-1)^4*(4*x-1)^4). - Alois P. Heinz, Sep 24 2012

Comment corrected by W. Edwin Clark, Sep 24 2012

cp's OEIS Frontend

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.

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A055602 Number of n X n binary matrices with no 0 rows or columns and with n+1 1's.

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A084485 Number of 3 X n 0-1 matrices which have n+2 1's and have no zero rows or zero columns.

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A084486 Number of 4 X n 0-1 matrices which have n+3 1's and have no zero rows or zero columns.

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