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A057150

Triangle read by rows: T(n,k) = number of k X k binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 5, 2, 1, 0, 0, 4, 11, 2, 1, 0, 0, 3, 21, 14, 2, 1, 0, 0, 1, 34, 49, 15, 2, 1, 0, 0, 1, 33, 131, 69, 15, 2, 1, 0, 0, 0, 33, 248, 288, 79, 15, 2, 1, 0, 0, 0, 19, 410, 840, 420, 82, 15, 2, 1, 0, 0, 0, 14, 531, 2144, 1744, 497, 83, 15, 2, 1

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OFFSET

1,9

COMMENTS

Also the number of non-isomorphic set multipartitions (multisets of sets) of weight n with k parts and k vertices. - Gus Wiseman, Nov 14 2018

LINKS

Table of n, a(n) for n=1..78.

EXAMPLE

[1], [0,1], [0,1,1], [0,1,2,1], [0,0,5,2,1], [0,0,4,11,2,1], ...;

There are 8 square binary matrices with 5 ones, with no zero rows or columns, up to row and column permutation: 5 of size 3 X 3:

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

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

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

2 of size 4 X 4:

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

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

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

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

and 1 of size 5 X 5:

[0 0 0 0 1]

[0 0 0 1 0]

[0 0 1 0 0]

[0 1 0 0 0]

[1 0 0 0 0].

From Gus Wiseman, Nov 14 2018: (Start)

Triangle begins:

0 1

0 1 1

0 1 2 1

0 0 5 2 1

0 0 4 11 2 1

0 0 3 21 14 2 1

0 0 1 34 49 15 2 1

0 0 1 33 131 69 15 2 1

0 0 0 33 248 288 79 15 2 1

Non-isomorphic representatives of the multiset partitions counted in row 6 {0,0,4,11,2,1} are:

{{12}{13}{23}} {{1}{1}{1}{234}} {{1}{2}{3}{3}{45}} {{1}{2}{3}{4}{5}{6}}

{{1}{23}{123}} {{1}{1}{24}{34}} {{1}{2}{3}{5}{45}}

{{13}{23}{23}} {{1}{1}{4}{234}}

{{3}{23}{123}} {{1}{2}{34}{34}}

{{1}{3}{24}{34}}

{{1}{3}{4}{234}}

{{1}{4}{24}{34}}

{{1}{4}{4}{234}}

{{2}{4}{12}{34}}

{{3}{4}{12}{34}}

{{4}{4}{12}{34}}

(End)

MATHEMATICA

permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];

c[p_List, q_List, k_] := SeriesCoefficient[Product[Product[(1 + O[x]^(k + 1) + x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}], {i, 1, Length[p]}], {x, 0, k}];

M[m_, n_, k_] := M[m, n, k] = Module[{s = 0}, Do[Do[s += permcount[p]* permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];

T[n_, k_] := M[k, k, n] - 2*M[k, k - 1, n] + M[k - 1, k - 1, n];

Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 10 2019, after Andrew Howroyd *)

PROG

(PARI) \\ See A321609 for M.

T(n, k) = M(k, k, n) - 2*M(k, k-1, n) + M(k-1, k-1, n); \\ Andrew Howroyd, Nov 14 2018

CROSSREFS

Row sums give A057151.

Cf. A049311, A056037, A056079, A056080, A057149, A057151, A057152.