A344112 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not functions.

1, 3, 12, 7, 56, 485, 15, 240, 4015, 65280, 31, 992, 32525, 1047552, 33551307, 63, 4032, 261415, 16773120, 1073726199, 68719430080, 127, 16256, 2094965, 268419072, 34359660243, 4398046231168, 562949952597769, 255, 65280, 16770655, 4294901760, 1099511237151

Offset: 1

Author

Mohammad K. Azarian, May 10 2021

Keywords

• easy
• nonn
• tabl

Examples

			T(2,2) = (number of relations) - (number of functions) = 2^4 - 4 = 12.
Triangle T(n,k) begins:
   1;
   3,  12;
   7,  56,   485;
  15, 240,  4015,   65280;
  31, 992, 32525, 1047552, 33551307;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1275 (rows n = 1..50, flattened)
• Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Crossrefs

Cf. A101030, A199656, A036679, A344110.

Programs

Mathematica

Column[Table[2^(n*k) - k^n, {n, 10}, {k, n}], Center]

Formula

T(n,k) = 2^(n*k) - k^n, n,k >= 1.