A344110 Triangle read by rows: T(n,k) = 2^(n*k), n >= 0, 0 <= k <= n.

1, 1, 2, 1, 4, 16, 1, 8, 64, 512, 1, 16, 256, 4096, 65536, 1, 32, 1024, 32768, 1048576, 33554432, 1, 64, 4096, 262144, 16777216, 1073741824, 68719476736, 1, 128, 16384, 2097152, 268435456, 34359738368, 4398046511104, 562949953421312

Offset: 0

Author

Mohammad K. Azarian, May 10 2021

Keywords

• easy
• nonn
• tabl

Comments

T(n, k) is the number of relations from an n-element set into a k-element set, n >= 0, 0 <= k <= n.

T(n,k) is the size of the right principal ideal generated by A where A is an n X n matrix over GF(2) having rank k. The right principal ideal of A contains precisely the matrices whose image is contained in the image of A. - Geoffrey Critzer, Sep 25 2022

Examples

			T(3,3) = number of relations from a 3-element set into a 3-element set=2^(3*3)=512.
Triangle begins:
   1
   1   2
   1   4      16
   1   8      64      512
   1  16     256     4096      65536
   1  32    1024    32768    1048576    33554432
   ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1325 (rows n = 0..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. A000079, A089072, A008292, A031971, A117401, A344260 (row sums).

Cf. A022166, A288853.

Programs

Mathematica

Table[2^(n*k), {n, 0, 10}, {k, 0, n}]

Formula

T(n,k) = 2^(n*k).

T(n,k) = Sum_{j=0..k} A288853(n,j)*A022166(n,j). - Geoffrey Critzer, Jan 02 2023