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

%I A089072 #60 Jun 23 2024 11:51:55
%S A089072 1,1,4,1,8,27,1,16,81,256,1,32,243,1024,3125,1,64,729,4096,15625,
%T A089072 46656,1,128,2187,16384,78125,279936,823543,1,256,6561,65536,390625,
%U A089072 1679616,5764801,16777216,1,512,19683,262144,1953125,10077696,40353607,134217728,387420489
%N A089072 Triangle read by rows: T(n,k) = k^n, n >= 1, 1 <= k <= n.
%C A089072 T(n, k) = number of mappings from an n-element set into a k-element set. - _Clark Kimberling_, Nov 26 2004
%C A089072 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
%C A089072 From _Manfred Boergens_, Jun 23 2024: (Start)
%C A089072 In the following two comments the restriction k<=n can be lifted, allowing all k>=1.
%C A089072 T(n,k) is the number of n X k binary matrices with row sums = 1.
%C A089072 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.
%C A089072 For nonempty A_j see A019538.
%C A089072 For tuples with "disjoint" dropped see A092477.
%C A089072 For tuples with nonempty A_j and with "disjoint" dropped see A218695. (End)
%H A089072 Reinhard Zumkeller, <a href="/A089072/b089072.txt">Rows n = 1..100 of triangle, flattened</a>
%H A089072 Mohammad K. Azarian, <a href="https://doi.org/10.12988/imf.2022.912321">Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions</a>, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Theorem 2.1(ii).
%F A089072 Sum_{k=1..n} T(n, k) = A031971(n).
%F A089072 T(n, n) = A000312(n).
%F A089072 T(2*n, n) = A062206(n).
%F A089072 a(n) = (n + T*(1-T)/2)^T, where T = round(sqrt(2*n),0). - _Gerald Hillier_, Apr 12 2015
%F A089072 T(n,k) = A051129(n,k). - _R. J. Mathar_, Dec 10 2015
%F A089072 T(n,k) = Sum_{i=0..k} Stirling2(n,i)*binomial(k,i)*i!. - _Geoffrey Critzer_, Dec 30 2021
%F A089072 From _G. C. Greubel_, Nov 01 2022: (Start)
%F A089072 T(n, n-1) = A007778(n-1), n >= 2.
%F A089072 T(n, n-2) = A008788(n-2), n >= 3.
%F A089072 T(2*n+1, n) = A085526(n).
%F A089072 T(2*n-1, n) = A085524(n).
%F A089072 T(2*n-1, n-1) = A085526(n-1), n >= 2.
%F A089072 T(3*n, n) = A083282(n).
%F A089072 Sum_{k=1..n} (-1)^k * T(n, k) = (-1)^n * A120485(n).
%F A089072 Sum_{k=1..floor(n/2)} T(n-k, k) = A226065(n).
%F A089072 Sum_{k=1..floor(n/2)} T(n, k) = A352981(n).
%F A089072 Sum_{k=1..floor(n/3)} T(n, k) = A352982(n). (End)
%e A089072 Triangle begins:
%e A089072   1;
%e A089072   1,  4;
%e A089072   1,  8,  27;
%e A089072   1, 16,  81,  256;
%e A089072   1, 32, 243, 1024,  3125;
%e A089072   1, 64, 729, 4096, 15625, 46656;
%e A089072   ...
%t A089072 Column[Table[k^n, {n, 8}, {k, n}], Center] (* _Alonso del Arte_, Nov 14 2011 *)
%o A089072 (Haskell)
%o A089072 a089072 = flip (^)
%o A089072 a089072_row n = map (a089072 n) [1..n]
%o A089072 a089072_tabl = map a089072_row [1..]  -- _Reinhard Zumkeller_, Mar 18 2013
%o A089072 (Magma) [k^n: k in [1..n], n in [1..12]]; // _G. C. Greubel_, Nov 01 2022
%o A089072 (SageMath) flatten([[k^n for k in range(1,n+1)] for n in range(1,12)]) # _G. C. Greubel_, Nov 01 2022
%Y A089072 Related to triangle of Eulerian numbers A008292.
%Y A089072 Cf. A000312, A007778, A008788, A031971 (row sums), A062206, A083282.
%Y A089072 Cf. A085524, A085526, A120485, A226065, A352981, A252982.
%Y A089072 Cf. A019538, A092477, A218695.
%K A089072 easy,nonn,tabl
%O A089072 1,3
%A A089072 _Alford Arnold_, Dec 04 2003
%E A089072 More terms and better definition from Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004
%E A089072 Offset corrected by _Reinhard Zumkeller_, Mar 18 2013