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A370505 T(n,k) is the difference between the number of k-dist-increasing and (k-1)-dist-increasing permutations of [n], where p is k-dist-increasing if k>=0 and p(i) for all i in [n-k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A370505 #36 Feb 27 2024 15:39:44
%S A370505 1,0,1,0,1,1,0,1,2,3,0,1,5,6,12,0,1,9,20,30,60,0,1,19,70,90,180,360,0,
%T A370505 1,34,175,420,630,1260,2520,0,1,69,490,1960,2520,5040,10080,20160,0,1,
%U A370505 125,1554,5880,15120,22680,45360,90720,181440,0,1,251,3948,21000,88200,113400,226800,453600,907200,1814400
%N A370505 T(n,k) is the difference between the number of k-dist-increasing and (k-1)-dist-increasing permutations of [n], where p is k-dist-increasing if k>=0 and p(i)<p(i+k) for all i in [n-k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A370505 Alois P. Heinz, <a href="/A370505/b370505.txt">Rows n = 0..150, flattened</a>
%H A370505 Wikipedia, <a href="https://en.wikipedia.org/wiki/K-sorted_sequence">K-sorted sequence</a>
%H A370505 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A370505 T(n,k) = A248686(n,k) - A248686(n,k-1) for k>=2.
%F A370505 Sum_{k=0..n} (1+n-k) * T(n,k) = A248687(n) for n>=1.
%e A370505 T(0,0) = 1: (only) the empty permutation is 0-dist-increasing.
%e A370505 T(4,2) = 5 = 6 - 1 = |{1234, 1243, 1324, 2134, 2143, 3142}| - |{1234}|.
%e A370505 Permutation 3142 is 2-dist-increasing and 4-dist-increasing but not 3-dist-increasing.
%e A370505 Triangle T(n,k) begins:
%e A370505   1;
%e A370505   0, 1;
%e A370505   0, 1,   1;
%e A370505   0, 1,   2,    3;
%e A370505   0, 1,   5,    6,   12;
%e A370505   0, 1,   9,   20,   30,    60;
%e A370505   0, 1,  19,   70,   90,   180,   360;
%e A370505   0, 1,  34,  175,  420,   630,  1260,  2520;
%e A370505   0, 1,  69,  490, 1960,  2520,  5040, 10080, 20160;
%e A370505   0, 1, 125, 1554, 5880, 15120, 22680, 45360, 90720, 181440;
%e A370505   ...
%p A370505 b:= proc(n, k) option remember; `if`(k<1,
%p A370505      `if`(n=k, 1, 0), n!/mul(iquo(n+i, k)!, i=0..k-1))
%p A370505     end:
%p A370505 T:= (n, k)-> b(n, k)-b(n, k-1):
%p A370505 seq(seq(T(n, k), k=0..n), n=0..10);
%Y A370505 Columns k=0-2 give: A000007, A057427, A014495.
%Y A370505 Row sums give A000142.
%Y A370505 Main diagonal gives A001710.
%Y A370505 T(2n,n+1) gives A000680 for n>=1.
%Y A370505 T(2n,n) gives A370576.
%Y A370505 Cf. A248686, A248687, A370506, A370507.
%K A370505 nonn,tabl
%O A370505 0,9
%A A370505 _Alois P. Heinz_, Feb 20 2024