A384105 Triangle read by rows: T(n,k) is the number of binary relations on a set of n objects, exactly k of which are self referencing, 0 <= k <= n.

1, 1, 1, 3, 4, 3, 16, 36, 36, 16, 218, 752, 1104, 752, 218, 9608, 45960, 90416, 90416, 45960, 9608, 1540944, 9133760, 22692704, 30194176, 22692704, 9133760, 1540944, 882033440, 6154473664, 18425858880, 30679088480, 30679088480, 18425858880, 6154473664, 882033440

Offset: 0

Author

Peter Dolland, May 19 2025

Keywords

• nonn
• tabl

Comments

Also the number of essentially different simple digraphs on a node set A of size n with a distinguished subset B of size k, where elements are indistinguishable within B and within A \ B.

Examples

			Triangle starts:
            1
            1,              1
            3,              4,              3
           16,             36,             36,              16
          218,            752,           1104,             752,             218
         9608,          45960,          90416,           90416,           45960, ...
      1540944,        9133760,       22692704,        30194176,        22692704, ...
    882033440,     6154473664,    18425858880,     30679088480,     30679088480, ...
1793359192848, 14334221970688, 50138592081152, 100240050239744, 125284653092864, ...
...
		

Links

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

Crossrefs

Cf. A000273 (edge cases), A000595 (row sums), A353996, A328874, A383617.

Formula

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

T(n,0) = T(n,n) = A000273(n).

T(n,1) = T(n,n-1) = A353996(n+1) = A329874(n,4).

Sum_{k=0..n} T(n,k) = A000595(n).