A089072 Triangle read by rows: T(n,k) = k^n, n >= 1, 1 <= k <= n.

1, 1, 4, 1, 8, 27, 1, 16, 81, 256, 1, 32, 243, 1024, 3125, 1, 64, 729, 4096, 15625, 46656, 1, 128, 2187, 16384, 78125, 279936, 823543, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489

Offset: 1

Author

Alford Arnold, Dec 04 2003

Keywords

• easy
• nonn
• tabl

Comments

T(n, k) = number of mappings from an n-element set into a k-element set. - Clark Kimberling, Nov 26 2004

Let S be the semigroup of (full) transformations on [n]. Let a be in S with rank(a) = k. Then T(n,k) = |a S|, the number of elements in the right principal ideal generated by a. - Geoffrey Critzer, Dec 30 2021

From Manfred Boergens, Jun 23 2024: (Start)

In the following two comments the restriction k<=n can be lifted, allowing all k>=1.

T(n,k) is the number of n X k binary matrices with row sums = 1.

T(n,k) is the number of coverings of [n] by tuples (A_1,...,A_k) in P([n])^k with disjoint A_j, with P(.) denoting the power set.

For nonempty A_j see A019538.

For tuples with "disjoint" dropped see A092477.

For tuples with nonempty A_j and with "disjoint" dropped see A218695. (End)

Examples

			Triangle begins:
  1;
  1,  4;
  1,  8,  27;
  1, 16,  81,  256;
  1, 32, 243, 1024,  3125;
  1, 64, 729, 4096, 15625, 46656;
  ...
		

Links

• Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
• Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Theorem 2.1(ii).

Crossrefs

Related to triangle of Eulerian numbers A008292.

Cf. A000312, A007778, A008788, A031971 (row sums), A062206, A083282.

Cf. A085524, A085526, A120485, A226065, A352981, A252982.

Cf. A019538, A092477, A218695.

Programs

Haskell

a089072 = flip (^)
a089072_row n = map (a089072 n) [1..n]
a089072_tabl = map a089072_row [1..]  -- Reinhard Zumkeller, Mar 18 2013

Magma

[k^n: k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 01 2022

Mathematica

Column[Table[k^n, {n, 8}, {k, n}], Center] (* Alonso del Arte, Nov 14 2011 *)

SageMath

flatten([[k^n for k in range(1,n+1)] for n in range(1,12)]) # G. C. Greubel, Nov 01 2022

Formula

Sum_{k=1..n} T(n, k) = A031971(n).

T(n, n) = A000312(n).

T(2*n, n) = A062206(n).

a(n) = (n + T*(1-T)/2)^T, where T = round(sqrt(2*n),0). - Gerald Hillier, Apr 12 2015

T(n,k) = A051129(n,k). - R. J. Mathar, Dec 10 2015

T(n,k) = Sum_{i=0..k} Stirling2(n,i)*binomial(k,i)*i!. - Geoffrey Critzer, Dec 30 2021

From G. C. Greubel, Nov 01 2022: (Start)

T(n, n-1) = A007778(n-1), n >= 2.

T(n, n-2) = A008788(n-2), n >= 3.

T(2*n+1, n) = A085526(n).

T(2*n-1, n) = A085524(n).

T(2*n-1, n-1) = A085526(n-1), n >= 2.

T(3*n, n) = A083282(n).

Sum_{k=1..n} (-1)^k * T(n, k) = (-1)^n * A120485(n).

Sum_{k=1..floor(n/2)} T(n-k, k) = A226065(n).

Sum_{k=1..floor(n/2)} T(n, k) = A352981(n).

Sum_{k=1..floor(n/3)} T(n, k) = A352982(n). (End)

Extensions

More terms and better definition from Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004

Offset corrected by Reinhard Zumkeller, Mar 18 2013