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A370506
T(n,k) is the number permutations p of [n] that are j-dist-increasing for exactly k distinct values j in [n], where p is j-dist-increasing if j>=0 and p(i)<p(i+j) for all i in [n-j]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
3
1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 11, 8, 4, 1, 0, 55, 38, 19, 7, 1, 0, 319, 228, 110, 50, 12, 1, 0, 2233, 1574, 775, 322, 115, 20, 1, 0, 17641, 12524, 6216, 2611, 1033, 261, 33, 1, 0, 158769, 112084, 55692, 23585, 9103, 3006, 586, 54, 1, 0, 1578667, 1119496, 556754, 238425, 91764, 33929, 8372, 1304, 88, 1
OFFSET
0,8
FORMULA
Sum_{k=0..n} k * T(n,k) = A248687(n) for n>=1.
EXAMPLE
T(4,1) = 11: 2341, 2431, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321.
T(4,2) = 8: 1342, 1423, 1432, 2314, 2413, 3124, 3142, 3214.
T(4,3) = 4: 1243, 1324, 2134, 2143.
T(4,4) = 1: 1234.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 3, 2, 1;
0, 11, 8, 4, 1;
0, 55, 38, 19, 7, 1;
0, 319, 228, 110, 50, 12, 1;
0, 2233, 1574, 775, 322, 115, 20, 1;
0, 17641, 12524, 6216, 2611, 1033, 261, 33, 1;
0, 158769, 112084, 55692, 23585, 9103, 3006, 586, 54, 1;
...
MAPLE
q:= proc(l, k) local i; for i from 1 to nops(l)-k do
if l[i]>=l[i+k] then return 0 fi od; 1
end:
b:= proc(n) option remember; add(x^add(
q(l, j), j=1..n), l=combinat[permute](n))
end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=0..n), n=0..8);
MATHEMATICA
q[l_, k_] := Module[{i}, For[i = 1, i <= Length[l]-k, i++,
If[l[[i]] >= l[[i+k]], Return@0]]; 1];
b[n_] := b[n] = Sum[x^Sum[q[l, j], {j, 1, n}], {l, Permutations[Range[n]]}];
T[n_, k_] := Coefficient[b[n], x, k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 24 2024, after Alois P. Heinz *)
CROSSREFS
Column k=0 gives A000007.
Column k=1 gives A370514 or A370507(n,n) for n>=1.
Row sums give A000142.
T(n,n-1) gives A000071(n+1).
Sequence in context: A357079 A259790 A246654 * A184182 A085771 A253286
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Feb 20 2024
STATUS
approved