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A188392 T(n,k) = number of (n*k) X k binary arrays with rows in nonincreasing order and n ones in every column.

%I A188392 #20 May 18 2020 03:00:59
%S A188392 1,2,1,5,3,1,15,16,4,1,52,139,39,5,1,203,1750,862,81,6,1,877,29388,
%T A188392 35775,4079,150,7,1,4140,624889,2406208,507549,15791,256,8,1,21147,
%U A188392 16255738,238773109,127126912,5442547,52450,410,9,1,115975,504717929,32867762616
%N A188392 T(n,k) = number of (n*k) X k binary arrays with rows in nonincreasing order and n ones in every column.
%H A188392 Andrew Howroyd, <a href="/A188392/b188392.txt">Table of n, a(n) for n = 1..181</a> (terms 1..69 from R. H. Hardin)
%e A188392 Array begins:
%e A188392 ========================================================================
%e A188392 n\k| 1  2   3      4         5            6            7               8
%e A188392 ---+--------------------------------------------------------------------
%e A188392 1  | 1  2   5     15        52          203           877           4140
%e A188392 2  | 1  3  16    139      1750        29388        624889       16255738
%e A188392 3  | 1  4  39    862     35775      2406208     238773109    32867762616
%e A188392 4  | 1  5  81   4079    507549    127126912   55643064708 38715666455777
%e A188392 5  | 1  6 150  15791   5442547   4762077620 8738543204786
%e A188392 6  | 1  7 256  52450  46757209 135029200594
%e A188392 7  | 1  8 410 154279 335279744
%e A188392 8  | 1  9 625 411180
%e A188392 9  | 1 10 915
%e A188392      ...
%e A188392 All solutions for 6 X 2
%e A188392 ..1..1....1..1....1..0....1..1
%e A188392 ..1..1....1..1....1..0....1..0
%e A188392 ..1..0....1..1....1..0....1..0
%e A188392 ..0..1....0..0....0..1....0..1
%e A188392 ..0..0....0..0....0..1....0..1
%e A188392 ..0..0....0..0....0..1....0..0
%o A188392 (PARI)
%o A188392 WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o A188392 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]!)}
%o A188392 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, 1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p,n,k), [1,n]); s*q[#q-j]))} \\ _Andrew Howroyd_, Dec 12 2018
%Y A188392 Rows 1..8 are A000110, A020554, A165434, A165435, A165436, A165437, A188393, A188394.
%Y A188392 Columns 3..7 are A011863(n+1), A175707, A188389, A188390, A188391.
%Y A188392 Main diagonal gives A188388.
%Y A188392 Cf. A188445, A219727, A330942.
%K A188392 nonn,tabl
%O A188392 1,2
%A A188392 _R. H. Hardin_, Mar 30 2011