A320600 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.

1, 2, 4, 4, 32, 156, 564, 1386, 1764

Offset

1,2

Comments

a(n) is even, because if a permutation is minimal, then so is its reverse.

The minimal sum is given by A327672. - Peter Kagey, Sep 21 2019

Links

Table of n, a(n) for n=1..9.

Sung Soo Kim, Problems and Solutions, Mathematics Magazine, 91:4 (2018), 310.

Michael Reid, Problems and Solutions, Mathematics Magazine, 92:4 (2019), 314.

Example

For n = 4 the a(4) = 4 permutations are
w_1 = (2,1,4,3),
w_2 = (2,4,1,3),
w_3 = (3,1,4,2), and
w_4 = (3,4,1,2).
In each case, f(1,w_i) + f(2,w_i) + f(3,w_i) + f(4,w_i) = A327672(4) = 7.

Crossrefs

Sequence in context: A092524 A137787 A225171 * A360685 A290606 A155952

Adjacent sequences: A320597 A320598 A320599 * A320601 A320602 A320603

Keyword

nonn,more

Author

Peter Kagey, Oct 16 2018

Status

approved