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A344115 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not one-to-one functions.

%I A344115 #30 Aug 26 2022 11:23:11
%S A344115 1,2,14,5,58,506,12,244,4072,65512,27,1004,32708,1048456,33554312,58,
%T A344115 4066,262024,16776856,1073741104,68719476016,121,16342,2096942,
%U A344115 268434616,34359735848,4398046506064,562949953416272,248,65480,16776880,4294965616,1099511621056
%N A344115 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not one-to-one functions.
%C A344115 If n=k, then T(n,k) = 2^(n^2) - n!, which is A344114, and if k<n, then T(n,k) = 2^(n*k), which is A344110.
%H A344115 Michael De Vlieger, <a href="/A344115/b344115.txt">Table of n, a(n) for n = 1..1275</a> (rows n = 1..50, flattened)
%H A344115 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 A344115 T(n,k) = 2^(n*k) - k!/(k-n)!, k >= n.
%e A344115 For T(2,2): the number of relations is 2^4 and the number of one-to-one functions is 2, so 2^4 - 2 = 14 and thus T(2,2) = 14.
%e A344115 Triangle T(n,k) begins:
%e A344115    1;
%e A344115    2,   14;
%e A344115    5,   58,   506;
%e A344115   12,  244,  4072,   65512;
%e A344115   27, 1004, 32708, 1048456, 33554312;
%t A344115 Table[2^(n*k) - k!/(k - n)!, {k, 10}, {n, k}] // Flatten
%Y A344115 Cf. A000312, A002416, A036679, A068424, A101030, A199656, A344110, A344112, A344113, A344114.
%K A344115 easy,nonn,tabl
%O A344115 1,2
%A A344115 _Mohammad K. Azarian_, Jun 06 2021