A300699 Irregular triangle read by rows: T(n, k) = number of vertices with rank k in concertina n-cube.

1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 6, 6, 3, 1, 1, 4, 12, 18, 28, 24, 28, 18, 12, 4, 1, 1, 5, 20, 40, 80, 95, 150, 150, 150, 150, 95, 80, 40, 20, 5, 1, 1, 6, 30, 75, 180, 270, 506, 660, 840, 1080, 1035, 1035, 1080, 840, 660, 506, 270, 180, 75, 30, 6, 1, 1, 7, 42, 126, 350, 630, 1337, 2107, 3192, 4760

Offset: 0

Author

Tilman Piesk, Mar 11 2018

Keywords

• nonn
• tabf

Comments

n-place formulas in first-order logic like Ax Ey P(x, y) ordered by implication form a graded poset, and its Hasse diagram is the concertina n-cube.

Sum of row n is A000629(n), the number of vertices of a concertina n-cube.

The rows are palindromic. Their lengths are the central polygonal numbers A000124 = 1, 2, 4, 7, 11, 16, 22, ... That means after row 0 rows of even and odd length follow each other in pairs.

The central values are 1, (1), (2), 6, 24, (150), (1035), 9030, 88760, (1002204), ... (Values next to the center in rows of even length are in parentheses.)

Maximal values are 1, 1, 2, 6, 28, 150, 1080, 9030, 88760, 1002204, ...

A300695 is a triangle of the same shape that shows the number of ranks in cocoon concertina hypercubes.

Examples

			First rows of the triangle:
    k   0   1   2   3   4   5    6    7    8    9   10  11  12  13  14  15
  n
  0     1
  1     1   1
  2     1   2   2   1
  3     1   3   6   6   6   3    1
  4     1   4  12  18  28  24   28   18   12    4    1
  5     1   5  20  40  80  95  150  150  150  150   95  80  40  20   5   1
  6     1   6  30  75 180 270  506  660  840 1080 1035 ...
The ranks of vertices of a concertina cube (n=3) can be seen in the linked Hasse diagrams. T(3, 4) = 6, so there are 6 vertices with rank 4.
Ey Ez Ax P(x, y, z) implies Ey Ax Ez P(x, y, z), and their ranks are 3 and 4. As the difference in rank is 1, this implication is an edge in the Hasse diagram.
		

Links

• Tilman Piesk, Rows 0..9, flattened
• Tilman Piesk, Rows 0..9
• Tilman Piesk, Formulas in predicate logic (Wikiversity)
• Tilman Piesk, Concertina cube Hasse diagram with labels and with highlighted ranks
• Tilman Piesk, Lists of vertices ordered by rank for n=2..6
• Tilman Piesk, Python code used to generate the sequence

Crossrefs

Cf. A000124, A000629, A300695.