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%I #28 May 15 2017 14:45:21
%S 2,10,8,28,54,22,60,190,204,52,110,490,916,676,114,182,1050,2878,3932,
%T 2118,240,280,1988,7278,15210,16148,6474,494,408,3444,15890,45738,
%U 77470,65210,19576,1004,570,5580,31192,115808,278358,389640,261708,58920,2026,770,8580,56484,258720,820118,1677048,1951700,1048008,176994,4072,1012,12650,96006,525444,2090296,5758802,10073698,9763628,4193580,531262,8166
%N Number of non-monotonic functions from [k] to [n-k].
%C Triangle T(n,k), 3<=k<=n-2, given by (n-k)^k-2*C(n-1,k)+(n-k) is derived using inclusion/exclusion. The triangle contains several other listed sequences: T(2n,n) is sequence A056174(n), number of monotonic functions from [n] to [n]; T(n+2,n) is sequence A005803(n), second-order Eulerian numbers; and T(n,3) is A006331(n-4), maximum accumulated number of electrons at energy level n.
%H Reinhard Zumkeller, <a href="/A189711/b189711.txt">Rows n=5..100 of triangle, flattened</a>
%H Dennis Walsh, <a href="http://frank.mtsu.edu/~dwalsh/MONOFUNC.pdf">Notes on finite monotonic and non-monotonic functions</a>
%F T(n,k)=(n-k)^k-2*C(n-1,k)+(n-k).
%F T(n,3) = A006331(n-4) for n>=5.
%F T(n+2,n) = A005803(n) for n>=3.
%F T(2n,n) = A056174(n) for n>=3.
%e Triangle T(n,k) begins
%e n\k 3 4 5 6 7 8 9
%e 5 2
%e 6 10 8
%e 7 28 54 22
%e 8 60 190 204 52
%e 9 110 490 916 676 114
%e 10 182 1050 2878 3932 2118 240
%e 11 280 1988 7278 15210 16148 6474 494
%e ...
%e For n=6 and k=4, T(6,4)=8 since there are 8 non-monotonic functions f from [4] to [2], namely, f = <f(1),f(2),f(3),f(4)> given by <1,1,2,1>, <1,2,1,1>, <1,2,2,1>, <1,2,1,2>, <2,2,1,2>, <2,1,2,2>, <2,1,1,2>, and <2,1,2,1>.
%p seq(seq((n-k)^k-2*binomial(n-1,k)+(n-k),k=3..(n-2)),n=5..15);
%t nmax = 15; t[n_, k_] := (n-k)^k-2*Binomial[n-1, k]+(n-k); Flatten[ Table[ t[n, k], {n, 5, nmax}, {k, 3, n-2}]](* _Jean-François Alcover_, Nov 18 2011, after Maple *)
%o (Haskell)
%o a189711 n k = (n - k) ^ k - 2 * a007318 (n - 1) k + n - k
%o a189711_row n = map (a189711 n) [3..n-2]
%o a189711_tabl = map a189711_row [5..]
%o -- _Reinhard Zumkeller_, May 16 2014
%Y Cf. A007318.
%K nonn,easy,nice,tabl
%O 5,1
%A _Dennis P. Walsh_, Apr 25 2011
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