A357213 Triangular array read by rows: T(n, k) = number of subsets s of {1, 2, ..., n} such max(s) - min(s) = k, for n >= 1, 0 <= k <= n-1.

1, 2, 1, 3, 2, 2, 4, 3, 4, 4, 5, 4, 6, 8, 8, 6, 5, 8, 12, 16, 16, 7, 6, 10, 16, 24, 32, 32, 8, 7, 12, 20, 32, 48, 64, 64, 9, 8, 14, 24, 40, 64, 96, 128, 128, 10, 9, 16, 28, 48, 80, 128, 192, 256, 256, 11, 10, 18, 32, 56, 96, 160, 256, 384, 512, 512, 12, 11

Offset: 1

Author

Clark Kimberling, Sep 24 2022

Keywords

• nonn
• tabl

Examples

			First 7 rows:
  1
  2     1
  3     2     2
  4     3     4     4
  5     4     6     8    8
  6     5     8    12   16    16
  7     6    10    16   24    32   32
		

Crossrefs

Cf. A000027, A130128 (obtained by deleting the first column), A000225 (row sums).

Programs

Mathematica

s[n_] := s[n] = Subsets[Range[n]]
u[n_, k_] := u[n, k] = Max[s[n][[k]]] - Min[s[n][[k]]]
v[n_] := Table[u[n, k], {k, 1, 2^n}];
t = Table[Count[v[n], i], {n, 1, 14}, {i, 0, n - 1}]
TableForm[t] (* A357213, array *)
Flatten[t]   (* A357213, sequence *)

PARI

T(n, k) = my(nb=0); forsubset(n, s, if (#s && (vecmax(s)-vecmin(s) == k), nb++)); nb; \\ Michel Marcus, Sep 26 2022

Formula

The n-th diagonal starts with n, followed by n*A000079(k), for k >= 0.

The columns, excluding the first, are given as in A130128 by T(n,k) = (n-k+1)*2^(k-1), for n >= 1, k >= 1.