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A321315 Number of permutations of [n] where the length of the longest increasing subsequence is larger than or equal to the length of the longest decreasing subsequence.

%I A321315 #9 Nov 04 2018 11:04:17
%S A321315 1,1,5,14,78,488,3161,25092,231428,2299664,24809824,296046900,
%T A321315 3863542365,54081895706,806425921874,12828011279528,217574673205512,
%U A321315 3914918953508792,74300528009315864,1482219340166034896,31035891175182089248,681299189806864371412,15649118660372502746968
%N A321315 Number of permutations of [n] where the length of the longest increasing subsequence is larger than or equal to the length of the longest decreasing subsequence.
%H A321315 Alois P. Heinz, <a href="/A321315/b321315.txt">Table of n, a(n) for n = 1..80</a>
%H A321315 Wikipedia, <a href="https://en.wikipedia.org/wiki/Longest_increasing_subsequence">Longest increasing subsequence</a>
%F A321315 a(n) = Sum_{k=0..n-1} A321316(n,k).
%F A321315 a(n) = A321313(n) + A321314(n).
%p A321315 h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
%p A321315     l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
%p A321315 f:= l-> `if`(l[1]>=nops(l), h(l)^2, 0):
%p A321315 g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
%p A321315      g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
%p A321315 a:= n-> g(n$2, []):
%p A321315 seq(a(n), n=1..23);
%Y A321315 Cf. A003316, A321313, A321314, A321316.
%K A321315 nonn
%O A321315 1,3
%A A321315 _Alois P. Heinz_, Nov 03 2018