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A370507 T(n,k) is the number permutations p of [n] that are k-dist-increasing but not j-dist-increasing for any j where p is j-dist-increasing if j>=0 and p(i) for all i in [n-j]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A370507 #28 Feb 29 2024 10:20:32
%S A370507 1,0,1,0,1,1,0,1,2,3,0,1,5,7,11,0,1,9,22,33,55,0,1,19,77,112,192,319,
%T A370507 0,1,34,189,480,788,1315,2233,0,1,69,526,2187,3500,5987,10409,17641,0,
%U A370507 1,125,1625,6811,18273,30568,53791,92917,158769,0,1,251,4111,23507,101424,167480,299769,528253,925337,1578667
%N A370507 T(n,k) is the number permutations p of [n] that are k-dist-increasing but not j-dist-increasing for any j<k, 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.
%H A370507 Wikipedia, <a href="https://en.wikipedia.org/wiki/K-sorted_sequence">K-sorted sequence</a>
%H A370507 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A370507 T(4,1) = 1: 1234.
%e A370507 T(4,2) = 5: 1243, 1324, 2134, 2143, 3142.
%e A370507 T(4,3) = 7: 1342, 1423, 1432, 2314, 2413, 3124, 3214.
%e A370507 T(4,4) = 11: 2341, 2431, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321.
%e A370507 Triangle T(n,k) begins:
%e A370507   1;
%e A370507   0, 1;
%e A370507   0, 1,   1;
%e A370507   0, 1,   2,    3;
%e A370507   0, 1,   5,    7,   11;
%e A370507   0, 1,   9,   22,   33,    55;
%e A370507   0, 1,  19,   77,  112,   192,   319;
%e A370507   0, 1,  34,  189,  480,   788,  1315,  2233;
%e A370507   0, 1,  69,  526, 2187,  3500,  5987, 10409, 17641;
%e A370507   0, 1, 125, 1625, 6811, 18273, 30568, 53791, 92917, 158769;
%e A370507   ...
%p A370507 q:= proc(l, k) local i; for i from 1 to nops(l)-k do
%p A370507       if l[i]>=l[i+k] then return 0 fi od; 1
%p A370507     end:
%p A370507 m:= proc(l) local k;
%p A370507       for k from 0 to nops(l) do if q(l, k)>0 then return k fi od
%p A370507     end:
%p A370507 b:= proc(n) b(n):= add(x^m(l), l=combinat[permute](n)) end:
%p A370507 T:= (n, k)-> coeff(b(n), x, k):
%p A370507 seq(seq(T(n, k), k=0..n), n=0..8);
%t A370507 q[l_, k_] := Module[{i}, For[i = 1, i <= Length[l] - k, i++, If[l[[i]] >= l[[i + k]], Return [0]]]; 1];
%t A370507 m[l_] := Module[{k}, For[k = 0, k <= Length[l], k++, If[q[l, k] > 0, Return[k]]]];
%t A370507 b[n_] := Sum[x^m[l], {l, Permutations[Range@n]}];
%t A370507 T[n_, k_] := Coefficient[b[n], x, k];
%t A370507 Table[Table[T[n, k], {k, 0, n}], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Feb 29 2024, after _Alois P. Heinz_ *)
%Y A370507 Columns k=0-2 give: A000007, A057427, A014495.
%Y A370507 Main diagonal gives A370514, also A370506(n,1) for n>=1.
%Y A370507 Row sums give A000142.
%Y A370507 Cf. A370505.
%K A370507 nonn,tabl
%O A370507 0,9
%A A370507 _Alois P. Heinz_, Feb 20 2024