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%I #18 Sep 21 2019 21:57:38
%S 1,2,4,4,32,156,564,1386,1764
%N Let w = (w_1, w_2, ..., w_n) be a permutation of the integers {1, 2, ..., n}, and let f(k, w) be the length of the longest monotone subsequence of (w_k, w_{k+1}, ..., w_n) starting with w_k. Then a(n) is the number of permutations w in S_n such that Sum_{k=1..n} f(k,w) is minimized.
%C a(n) is even, because if a permutation is minimal, then so is its reverse.
%C The minimal sum is given by A327672. - _Peter Kagey_, Sep 21 2019
%H Sung Soo Kim, <a href="https://doi.org/10.1080/0025570X.2018.1501259">Problems and Solutions</a>, Mathematics Magazine, 91:4 (2018), 310.
%H Michael Reid, <a href="https://www.tandfonline.com/doi/full/10.1080/0025570X.2019.1648111?af=R">Problems and Solutions</a>, Mathematics Magazine, 92:4 (2019), 314.
%e For n = 4 the a(4) = 4 permutations are
%e w_1 = (2,1,4,3),
%e w_2 = (2,4,1,3),
%e w_3 = (3,1,4,2), and
%e w_4 = (3,4,1,2).
%e In each case, f(1,w_i) + f(2,w_i) + f(3,w_i) + f(4,w_i) = A327672(4) = 7.
%K nonn,more
%O 1,2
%A _Peter Kagey_, Oct 16 2018
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