A321316
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.
Original entry on oeis.org
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, 256, 125, 81, 25, 0, 1, 1, 0, 36, 196, 421, 1225, 1282, 1225, 421, 196, 36, 0, 1, 1, 0, 49, 400, 1225, 4292, 9261, 9864, 9261, 4292, 1225, 400, 49, 0, 1
Offset: 1
: 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, 256, 125, 81, 25, 0, 1 ;
: 1, 0, 36, 196, 421, 1225, 1282, 1225, 421, 196, 36, 0, 1 ;
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h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> h(l)^2*x^(l[1]-nops(l)) :
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
b:= proc(n) option remember; g(n$2, []) end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=1-n..n-1), n=1..10);
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h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[j > l[[k]], 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
f[l_] := h[l]^2*x^(l[[1]] - Length[l]);
g[n_, i_, l_] := If[n == 0 || i == 1, f[Join[l, Table[1, {n}]]], g[n, i - 1, l] + g[n - i, Min[i, n - i], Append[l, i]]];
b[n_] := b[n] = g[n, n, {}];
T[n_, k_] := Coefficient[b[n], x, k];
Table[Table[T[n, k], {k, 1 - n, n - 1}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Feb 27 2021, after Alois P. Heinz *)
A321313
Number of permutations of [n] with equal lengths of the longest increasing subsequence and the longest decreasing subsequence.
Original entry on oeis.org
1, 0, 4, 4, 36, 256, 1282, 9864, 99976, 970528, 9702848, 113092200, 1500063930, 20985500212, 305177475748, 4733232671056, 79461918315024, 1427464201289584, 26955955609799728, 531536672155429792, 10980840178654738496, 238597651836121062824, 5446220581860028853936
Offset: 1
-
h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> `if`(l[1]=nops(l), h(l)^2, 0):
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);
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h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[j > l[[k]], 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
f[l_] := If[l[[1]] == Length[l], h[l]^2, 0];
g[n_, i_, l_] := If[n == 0 || i == 1, f[Join[l, Table[1, {n}]]], g[n, i - 1, l] + g[n - i, Min[i, n - i], Append[l, i]]];
a[n_] := g[n, n, {}];
Array[a, 25] (* Jean-François Alcover, Aug 31 2021, after Alois P. Heinz *)
A321314
Number of permutations of [n] where the length of the longest increasing subsequence is larger than the length of the longest decreasing subsequence.
Original entry on oeis.org
0, 1, 1, 10, 42, 232, 1879, 15228, 131452, 1329136, 15106976, 182954700, 2363478435, 33096395494, 501248446126, 8094778608472, 138112754890488, 2487454752219208, 47344572399516136, 950682668010605104, 20055050996527350752, 442701537970743308588, 10202898078512473893032
Offset: 1
-
h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> `if`(l[1] `if`(n=0 or i=1, f([l[], 1$n]),
g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);
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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}]];
f[l_] := If[l[[1]] < Length[l], h[l]^2, 0];
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]]];
a[n_] := g[n, n, {}];
Array[a, 25] (* Jean-François Alcover, Aug 31 2021, after Alois P. Heinz *)
Showing 1-3 of 3 results.