A321587 Number of (0,1)-matrices with n ones, no zero rows or columns, and distinct rows.

1, 1, 3, 17, 129, 1227, 14123, 190265, 2934359, 50975647, 984801759, 20941104299, 486007744671, 12223797601887, 331190083773701, 9616356919931711, 297887922137531747, 9805965265937326129, 341827167387114704421, 12579123760272833723975, 487315396984696657840761

Offset

0,3

Comments

Also number of colored compositions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and the patterns for parts i have i distinct colors in increasing order. a(3) = 17: 2ab1a, 2ab1b, 1a2ab, 1b2ab, 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a. - Alois P. Heinz, Sep 17 2019

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

a(n) ~ c * d^n * n!, where d = 1.938593839617140963759657977... and c = 0.350862127201784401195038... - Vaclav Kotesovec, Feb 05 2022

Example

The a(3) = 17 matrices:
  [1 1 1]
.
  [1 1] [1 1] [1 1 0] [1 0 1] [1 0] [1 0 0] [0 1 1] [0 1] [0 1 0] [0 0 1]
  [1 0] [0 1] [0 0 1] [0 1 0] [1 1] [0 1 1] [1 0 0] [1 1] [1 0 1] [1 1 0]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Maple

C:= binomial:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k, p+j)*C(C(k, i), j), j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
seq(a(n), n=0..21);  # Alois P. Heinz, Sep 16 2019

Mathematica

prs2mat[prs_]:=Table[Count[prs, {i, j}], {i, Union[First/@prs]}, {j, Union[Last/@prs]}];
Table[Length[Select[Subsets[Tuples[Range[n], 2], {n}], And[Union[First/@#]==Range[Max@@First/@#], Union[Last/@#]==Range[Max@@Last/@#], UnsameQ@@prs2mat[#]]&]], {n, 5}]

Crossrefs

Cf. A007716, A049311, A101370, A120733, A283877, A316980, A321446, A321515.

Row sums of A327583, A327584.

Sequence in context: A179300 A199493 A245059 * A089815 A328402 A247788

Adjacent sequences: A321584 A321585 A321586 * A321588 A321589 A321590

Keyword

nonn

Author

Gus Wiseman, Nov 13 2018

Extensions

a(7)-a(20) from Alois P. Heinz, Sep 16 2019

Status

approved