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

1,3

Comments

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.

Number of coverings of an n-element set by m nonempty subsets (not necessarily disjunct). For the disjunct case cf. A019538. (End)

Links

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.

Formula

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

Example

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;
  ...

Maple

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

Mathematica

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 *)

Prog

(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(); ); };
tabl(8); \\ Indranil Ghosh, Mar 14 2017
(Python)
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

Crossrefs

Cf. A058482 (this gives the general formula, but values only for m=3).

Main diagonal gives A048291.

Column 2 is A058481.

Cf. A019538.

Sequence in context: A064051 A147385 A147347 * A082172 A053288 A282917

Adjacent sequences: A183106 A183107 A183108 * A183110 A183111 A183112

Keyword

nonn,tabl

Author

Steffen Eger, Feb 01 2011

Status

approved