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Number T(n,k) of permutations of [n] whose difference between the length of the longest increasing subsequence and the length of the longest decreasing subsequence equals k; triangle T(n,k), n >= 1, 1-n <= k <= n-1, read by rows.
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%I #28 Feb 27 2021 11:18:31

%S 1,1,0,1,1,0,4,0,1,1,0,9,4,9,0,1,1,0,16,25,36,25,16,0,1,1,0,25,81,125,

%T 256,125,81,25,0,1,1,0,36,196,421,1225,1282,1225,421,196,36,0,1,1,0,

%U 49,400,1225,4292,9261,9864,9261,4292,1225,400,49,0,1

%N Number T(n,k) of permutations of [n] whose difference between the length of the longest increasing subsequence and the length of the longest decreasing subsequence equals k; triangle T(n,k), n >= 1, 1-n <= k <= n-1, read by rows.

%H Alois P. Heinz, <a href="/A321316/b321316.txt">Rows n = 1..80, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Longest_increasing_subsequence">Longest increasing subsequence</a>

%F T(n,k) = T(n,-k).

%F Sum_{k=1..n-1} T(n,k) = A321314(n).

%F Sum_{k=0..n-1} T(n,k) = A321315(n).

%F (1/2) * Sum_{k=1-n..n-1} abs(k) * T(n,k) = A321277(n).

%F (1/2) * Sum_{k=1-n..n-1} k^2 * T(n,k) = A321278(n).

%e : 1 ;

%e : 1, 0, 1 ;

%e : 1, 0, 4, 0, 1 ;

%e : 1, 0, 9, 4, 9, 0, 1 ;

%e : 1, 0, 16, 25, 36, 25, 16, 0, 1 ;

%e : 1, 0, 25, 81, 125, 256, 125, 81, 25, 0, 1 ;

%e : 1, 0, 36, 196, 421, 1225, 1282, 1225, 421, 196, 36, 0, 1 ;

%p h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>

%p l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):

%p f:= l-> h(l)^2*x^(l[1]-nops(l)) :

%p g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),

%p g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):

%p b:= proc(n) option remember; g(n$2, []) end:

%p T:= (n, k)-> coeff(b(n), x, k):

%p seq(seq(T(n, k), k=1-n..n-1), n=1..10);

%t h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[j > l[[k]], 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];

%t f[l_] := h[l]^2*x^(l[[1]] - Length[l]);

%t g[n_, i_, l_] := If[n == 0 || i == 1, f[Join[l, Table[1, {n}]]], g[n, i - 1, l] + g[n - i, Min[i, n - i], Append[l, i]]];

%t b[n_] := b[n] = g[n, n, {}];

%t T[n_, k_] := Coefficient[b[n], x, k];

%t Table[Table[T[n, k], {k, 1 - n, n - 1}], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Feb 27 2021, after _Alois P. Heinz_ *)

%Y Column k=0 gives A321313.

%Y Row sums give A000142.

%Y T(n+1,n-2) gives A000290.

%Y Cf. A303697, A321277, A321278, A321314, A321315.

%K nonn,tabf

%O 1,7

%A _Alois P. Heinz_, Nov 03 2018