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A334550 Triangle read by rows: T(n,k) is the number of binary matrices with n ones, k columns and no zero rows or columns, up to permutations of rows and columns. 2
1, 1, 2, 1, 2, 3, 1, 5, 5, 5, 1, 5, 12, 9, 7, 1, 9, 23, 29, 17, 11, 1, 9, 39, 62, 57, 28, 15, 1, 14, 63, 147, 154, 110, 47, 22, 1, 14, 102, 278, 409, 329, 194, 73, 30, 1, 20, 150, 568, 991, 1023, 664, 335, 114, 42, 1, 20, 221, 1020, 2334, 2844, 2267, 1243, 549, 170, 56 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

T(n,k) is also the number of non-isomorphic set multipartitions (multisets of sets) of weight n with k parts.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

EXAMPLE

Triangle begins:

  1;

  1,  2;

  1,  2,  3;

  1,  5,  5,   5;

  1,  5, 12,   9,   7;

  1,  9, 23,  29,  17,  11;

  1,  9, 39,  62,  57,  28, 15;

  1, 14, 63, 147, 154, 110, 47, 22;

  ...

The T(4,3) = 5 matrices are:

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

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

  [0 1 0]   [0 1 1]   [0 0 1]

  [0 0 1]

The T(4,3) = 5 the set multipartitions are:

  {{1,2}, {3}, {4}},

  {{1,2}, {3}, {3}},

  {{1,2}, {1}, {3}},

  {{1,2}, {1}, {1}},

  {{1,2}, {1}, {2}}.

PROG

(PARI) \\ See A321609 for definition of M.

T(n, k)={M(k, n, n) - M(k-1, n, n)}

for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print)

(PARI) \\ Faster version.

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

K(q, t, n)={prod(j=1, #q, (1+x^lcm(t, q[j]) + O(x*x^n))^gcd(t, q[j]))}

G(m, n)={my(s=0); forpart(q=m, s+=permcount(q)*exp(sum(t=1, n, (K(q, t, n)-1)/t) + O(x*x^n))); s/m!}

A(n, m=n)={my(p=sum(k=0, m, G(k, n)*y^k)*(1-y)); matrix(n, m, n, k, polcoef(polcoef(p, n, x), k, y))}

{ my(T=A(10)); for(n=1, #T, print(T[n, 1..n])) }

CROSSREFS

Row sums are A049311.

Main diagonal is A000041.

Cf. A317533, A321609.

Sequence in context: A078032 A162453 A008313 * A232177 A111377 A014046

Adjacent sequences:  A334547 A334548 A334549 * A334551 A334552 A334553

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Jul 03 2020

STATUS

approved

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Last modified March 3 14:58 EST 2021. Contains 341762 sequences. (Running on oeis4.)