A387672 Triangle read by rows: T(n,k) is the number of topological sequences of three dimensional lattice points with index n and depth k.

1, 0, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 12, 0, 0, 0, 1, 0, 23, 0, 0, 0, 0, 9, 0, 47, 0, 0, 0, 0, 3, 25, 0, 85, 0, 0, 0, 0, 0, 28, 61, 0, 159, 0, 0, 0, 0, 0, 12, 85, 125, 0, 281, 0, 0, 0, 0, 0, 5, 84, 203, 252, 0, 499, 0, 0, 0, 0, 0, 9, 48, 252, 441, 472, 0, 858

Offset

0,6

Comments

These sequences use the lexicographic well-ordering (i,j,k) < (x,y,z) and the partial order (i,j,k) <* (x,y,z) if i <= x, j <= y, and k <= z. A topological sequence S is an ordered collection of elements satisfying the following two criteria:

(1) For all b in S if a <* b then a is a element of S and appears before b.

(2) There is some element c not in S such that c < m, where m is the last element of S.

The depth of a topological sequence is the number of lattice points in the sequence. The index of a topological sequence S (x_1, x_2, ..., x_m) is the depth plus the sum of the indices of elements x_i of S such that x_i > x_{i+1}.

Links

Table of n, a(n) for n=0..77.

John Tyler Rascoe, Python code.

Example

Triangle begins:
      k=0  1  2  3   4   5   6    7    8    9    10
  n=0  [1]
  n=1  [0, 0]
  n=2  [0, 0, 2]
  n=3  [0, 0, 0, 5]
  n=4  [0, 0, 0, 0, 12]
  n=5  [0, 0, 0, 1,  0, 23]
  n=6  [0, 0, 0, 0,  9,  0, 47]
  n=7  [0, 0, 0, 0,  3, 25,  0,  85]
  n=8  [0, 0, 0, 0,  0, 28, 61,   0, 159]
  n=9  [0, 0, 0, 0,  0, 12, 85, 125,   0, 281]
  n=10 [0, 0, 0, 0,  0,  5, 84, 203, 252,   0, 499]
  ...
T(2,2) = 2: {(0,0,0), (0,1,0)}, {(0,0,0), (1,0,0)}.
T(3,3) = 5: {(0,0,0), (0,0,1), (0,1,0)}, {(0,0,0), (0,0,1), (1,0,0)}, {(0,0,0), (0,1,0), (0,2,0)}, {(0,0,0), (0,1,0), (1,0,0)}, {(0,0,0), (1,0,0), (2,0,0)}.
T(5,3) = 1: {(0,0,0), (1,0,0), (0,1,0)}.

Prog

(Python) # see links

Crossrefs

Cf. A000219, A000293, A002836 (row sums), A207542.

Sequence in context: A328820 A259863 A283666 * A131427 A153198 A182492

Adjacent sequences: A387669 A387670 A387671 * A387673 A387674 A387675

Keyword

nonn,tabl

Author

John Tyler Rascoe, Nov 30 2025

Status

approved