A331571 - OEIS

A331571

Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of distinct nonzero rows with n ones in every column and columns in nonincreasing lexicographic order.

1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 4, 3, 0, 1, 1, 8, 23, 0, 0, 1, 1, 16, 290, 184, 0, 0, 1, 1, 32, 4298, 17488, 840, 0, 0, 1, 1, 64, 79143, 2780752, 771305, 0, 0, 0, 1, 1, 128, 1702923, 689187720, 1496866413, 21770070, 0, 0, 0, 1, 1, 256, 42299820, 236477490418, 5261551562405, 585897733896, 328149360, 0, 0, 0, 1

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OFFSET

0,8

COMMENTS

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..209

FORMULA

A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A331567(n, j)/k!.

A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331569(n, j).

A(n, k) = 0 for k > 0, n > 2^(k-1).

A331653(n) = Sum_{d|n} A(n/d, d).

EXAMPLE

Array begins:

===============================================================

n\k | 0 1 2 3 4 5 6

----+----------------------------------------------------------

0 | 1 1 1 1 1 1 1 ...

1 | 1 1 2 4 8 16 32 ...

2 | 1 0 3 23 290 4298 79143 ...

3 | 1 0 0 184 17488 2780752 689187720 ...

4 | 1 0 0 840 771305 1496866413 5261551562405 ...

5 | 1 0 0 0 21770070 585897733896 30607728081550686 ...

6 | 1 0 0 0 328149360 161088785679360 ...

...

The A(2,2) = 3 matrices are:

[1 1] [1 0] [1 0]

[1 0] [1 1] [0 1]

[0 1] [0 1] [1 1]

PROG

(PARI)

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}

D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(WeighT(v)[n] + k - 1, k)/prod(i=1, #v, i^v[i]*v[i]!)}

T(n, k)={ my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, 1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }

CROSSREFS

Rows n=0..4 are A000012, A011782, A060090, A060491, A331652.

Cf. A330942, A331567, A331569, A331570, A331572, A331653.

Sequence in context: A102564 A215703 A292712 * A247504 A306800 A235955

Adjacent sequences: A331568 A331569 A331570 * A331572 A331573 A331574

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Jan 20 2020

STATUS

approved