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A347034 Triangle read by columns: T(n,k) is the number of functions from an n-element set to a k-element set that are not one-to-one, k>=n>=1.

%I A347034 #61 Jan 12 2023 06:32:16
%S A347034 0,0,2,0,3,21,0,4,40,232,0,5,65,505,3005,0,6,96,936,7056,45936,0,7,
%T A347034 133,1561,14287,112609,818503,0,8,176,2416,26048,241984,2056832,
%U A347034 16736896,0,9,225,3537,43929,470961,4601529,42683841,387057609,0,10,280,4960,69760,848800
%N A347034 Triangle read by columns: T(n,k) is the number of functions from an n-element set to a k-element set that are not one-to-one, k>=n>=1.
%C A347034 The formula for this sequence is Theorem 2.2(iv) of the author's paper, p. 131 (see the link).
%H A347034 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.
%F A347034 T(n,k) = k^n - k!/(k - n)!, k>=n.
%F A347034 T(n,n) = A036679(n).
%e A347034 For T(2,3): the number of functions is 3^2 and the number of one-to-one functions is 6, so 3^2 - 6 = 3 and thus T(2,3) = 3.
%e A347034 Triangle T(n,k) begins:
%e A347034        k=1  k=2   k=3   k=4    k=5     k=6
%e A347034   n=1:  0    0    0     0      0       0
%e A347034   n=2:       2    3     4      5       6
%e A347034   n=3:            21    40     65      96
%e A347034   n=4:                  232    505     936
%e A347034   n=5:                         3005    7056
%e A347034   n=6:                                 45936
%p A347034 A347034 := proc(n,k)
%p A347034     k^n-k!/(k-n)! ;
%p A347034 end proc:
%p A347034 seq(seq(A347034(n,k),n=1..k),k=1..12) ; # _R. J. Mathar_, Jan 12 2023
%t A347034 Table[k^n - k!/(k - n)!, {k, 12}, {n, k}] // Flatten
%o A347034 (PARI) T(n,k) = k^n - k!/(k - n)!;
%o A347034 row(k) = vector(k, i, T(i, k)); \\ _Michel Marcus_, Oct 01 2021
%Y A347034 Cf. A000312, A002416, A036679, A068424, A089072, A101030, A199656, A344110, A344112, A344113, A344114, A344115, A344116.
%K A347034 easy,nonn,tabl
%O A347034 1,3
%A A347034 _Mohammad K. Azarian_, Aug 28 2021