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%I #14 Dec 09 2025 13:23:13
%S 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,
%T 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,
%U 84,203,252,0,499,0,0,0,0,0,9,48,252,441,472,0,858
%N Triangle read by rows: T(n,k) is the number of topological sequences of three dimensional lattice points with index n and depth k.
%C 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:
%C (1) For all b in S if a <* b then a is a element of S and appears before b.
%C (2) There is some element c not in S such that c < m, where m is the last element of S.
%C 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}.
%H John Tyler Rascoe, <a href="/A387672/a387672.py.txt">Python code</a>.
%e Triangle begins:
%e k=0 1 2 3 4 5 6 7 8 9 10
%e n=0 [1]
%e n=1 [0, 0]
%e n=2 [0, 0, 2]
%e n=3 [0, 0, 0, 5]
%e n=4 [0, 0, 0, 0, 12]
%e n=5 [0, 0, 0, 1, 0, 23]
%e n=6 [0, 0, 0, 0, 9, 0, 47]
%e n=7 [0, 0, 0, 0, 3, 25, 0, 85]
%e n=8 [0, 0, 0, 0, 0, 28, 61, 0, 159]
%e n=9 [0, 0, 0, 0, 0, 12, 85, 125, 0, 281]
%e n=10 [0, 0, 0, 0, 0, 5, 84, 203, 252, 0, 499]
%e ...
%e T(2,2) = 2: {(0,0,0), (0,1,0)}, {(0,0,0), (1,0,0)}.
%e 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)}.
%e T(5,3) = 1: {(0,0,0), (1,0,0), (0,1,0)}.
%o (Python) # see links
%Y Cf. A000219, A000293, A002836 (row sums), A207542.
%K nonn,tabl
%O 0,6
%A _John Tyler Rascoe_, Nov 30 2025