A383617 Triangle read by rows: T(n,k) is the number of binary relations on a set of n objects, k of which are picked out, 0 <= k <= n.

1, 2, 2, 10, 16, 10, 104, 272, 272, 104, 3044, 11456, 16960, 11456, 3044, 291968, 1432608, 2842304, 2842304, 1432608, 291968, 96928992, 578431232, 1441700480, 1920352256, 1441700480, 578431232, 96928992, 112282908928, 784780122880, 2351993457920, 3918054495616, 3918054495616, 2351993457920, 784780122880, 112282908928

Offset: 0

Author

Peter Dolland, May 02 2025

Keywords

• nonn
• tabl

Comments

The row sums are the number of simple digraphs with n 4-colored nodes. The colors result from the four cases combining the property self-referencing (yes/no) with "picked out" (yes/no).

Examples

			Triangle starts:
            1;
            2,            2;
           10,           16,            10;
          104,          272,           272,           104;
         3044,        11456,         16960,         11456,          3044;
       291968,      1432608,       2842304,       2842304,       1432608,  291968;
     96928992,    578431232,    1441700480,    1920352256,    1441700480, ...
 112282908928, 784780122880, 2351993457920, 3918054495616, 3918054495616, ...
...
Example n=2, k=1: The both objects are differentiated. As a consequence all binary relations on two different objects have to be counted: These are the subsets of the cross product of the objects set with itself. This contains four pairs, so the number of subsets is 2^4 = 16.
		

Links

• Peter Dolland, Python calculation of T(n,k)

Crossrefs

Cf. A000595 (edge cases), A353996 (row sums), A329874 (4th column = row sums).

Formula

T(n,k) = T(n,n-k).

T(n,0) = T(n,n) = A000595(n).

Sum_{k=0..n} T(n,k) = A353996(n+1) = A329874(n,4).