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%I #67 Jan 15 2024 18:34:21
%S 1,1,7,1,25,265,1,79,2161,41503,1,241,16081,693601,24997921,1,727,
%T 115465,10924399,831719761,57366997447,1,2185,816985,167578321,
%U 26666530801,3776451407065,505874809287625
%N 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).
%C 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
%C From _Manfred Boergens_, Jul 25 2021: (Start)
%C The matrices in the definition are a superset of the matrices in the comment to A019538 by Manfred Boergens.
%C Number of coverings of an n-element set by m nonempty subsets (not necessarily disjunct). For the disjunct case cf. A019538. (End)
%H Indranil Ghosh, <a href="/A183109/b183109.txt">Rows 1..50, flattened</a>
%H Ch. A. Charalambides, <a href="https://doi.org/10.1016/0012-365X(79)90108-0">A problem of arrangements on chessboards and generalizations</a>, Discrete Mathematics 27.2 (1979): 179-186. (Generalizations.)
%H D. E. Knuth, <a href="http://dx.doi.org/10.2307/27642039">Problem 11243</a>, Am. Math. Montly 113 (8) (2006) page 759.
%H John Riordan and Paul R. Stein, <a href="https://doi.org/10.1016/0097-3165(72)90084-2">Arrangements on chessboards</a>, Journal of Combinatorial Theory, Series A 12.1 (1972): 72-80. See Table page 78.
%F T(n,m) = Sum_{j=0..m}(-1)^j*C(m, j)*(2^(m-j)-1)^n.
%F Recursion: T(m,n) = Sum_{k=1..m} T(k,n-1)*C(m,k)*2^k - T(m,n-1).
%F From _Robert FERREOL_, Mar 14 2017: (Start)
%F T(n,m) = Sum_{i = 0 .. n,j = 0 ..m}(-1)^(n+m+i+j)*C(n,i)*C(m,j)*2^(i*j).
%F Inverse formula of: 2^(n*m) = Sum_{i = 0 .. n , j = 0 ..m} C(n,i)*C(m,j)*T(i,j). (End)
%F A019538(j) <= a(j). - _Manfred Boergens_, Jul 25 2021
%e Triangle begins:
%e 1;
%e 1, 7;
%e 1, 25, 265;
%e 1, 79, 2161, 41503;
%e 1, 241, 16081, 693601, 24997921;
%e 1, 727, 115465, 10924399, 831719761, 57366997447;
%e 1, 2185, 816985, 167578321, 26666530801, 3776451407065, 505874809287625;
%e ...
%p A183109 := proc(n,m)
%p add((-1)^j*binomial(m,j)*(2^(m-j)-1)^n,j=0..m) ;
%p end proc:
%p seq(seq(A183109(n,m),m=1..n),n=1..10) ; # _R. J. Mathar_, Dec 03 2015
%t 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 *)
%o (PARI) 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(););};
%o tabl(8); \\ _Indranil Ghosh_, Mar 14 2017
%o (Python)
%o import math
%o f = math.factorial
%o def C(n,r): return f(n)//f(r)//f(n - r)
%o def T(n,m):
%o return sum((-1)**j*C(m,j)*(2**(m - j) - 1)**n for j in range (m+1))
%o i=1
%o for n in range(1,21):
%o for m in range(1, n+1):
%o print(str(i)+" "+str(T(n, m)))
%o i+=1 # _Indranil Ghosh_, Mar 14 2017
%Y Cf. A058482 (this gives the general formula, but values only for m=3).
%Y Main diagonal gives A048291.
%Y Column 2 is A058481.
%Y Cf. A019538.
%K nonn,tabl
%O 1,3
%A _Steffen Eger_, Feb 01 2011
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