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A101030 Triangle read by rows: T(n,k) = number of functions from an n-element set into but not onto a k-element set.

%I A101030 #20 Oct 17 2022 07:07:12
%S A101030 0,0,2,0,2,21,0,2,45,232,0,2,93,784,3005,0,2,189,2536,13825,45936,0,2,
%T A101030 381,7984,61325,264816,818503,0,2,765,24712,264625,1488096,5623681,
%U A101030 16736896,0,2,1533,75664,1119005,8172576,38025127,132766208,387057609,0
%N A101030 Triangle read by rows: T(n,k) = number of functions from an n-element set into but not onto a k-element set.
%H A101030 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 2.2(v).
%H A101030 D. P. Walsh, <a href="http://capone.mtsu.edu/dwalsh/NONSURJ.pdf">A note on non-surjective functions from [n] to [k]</a>.
%F A101030 T(n,k) = A089072(n,k) - A019538(n,k).
%F A101030 T(n,k) = Sum_{j=1..k} (-1)^(j-1)*C(k,j)*(k-j)^n. - _Dennis P. Walsh_, Apr 13 2016
%F A101030 T(n,k) = k^n - k!*Stirling2(n,k). - _Dennis P. Walsh_, Apr 13 2016
%e A101030 T(3,3) = #(functions into) - #(functions onto) = 3^3 - 6 = 21
%e A101030 Triangle T(n,k) begins:
%e A101030   0,
%e A101030   0, 2;
%e A101030   0, 2,   21;
%e A101030   0, 2,   45,   232;
%e A101030   0, 2,   93,   784,    3005;
%e A101030   0, 2,  189,  2536,   13825,   45936;
%e A101030   0, 2,  381,  7984,   61325,  264816,   818503;
%e A101030   0, 2,  765, 24712,  264625, 1488096,  5623681,  16736896;
%e A101030   0, 2, 1533, 75664, 1119005, 8172576, 38025127, 132766208, 387057609;
%p A101030 T:=(n, k)->sum((-1)^(j-1)*binomial(k, j)*(k-j)^n, j=1..k);
%p A101030 seq(seq(T(n, k), k=1..n), n=1..15); # _Dennis P. Walsh_, Apr 13 2016
%Y A101030 Cf. A199656, A036679 (diagonal).
%K A101030 nonn,tabl,easy
%O A101030 1,3
%A A101030 _Clark Kimberling_, Nov 26 2004
%E A101030 Offset corrected from 0 to 1 by _Dennis P. Walsh_, Apr 13 2016