A300693 -id:A300693 - cp's OEIS frontend

A300700 Triangle read by rows: T(n, n-k) = number of k-faces of the concertina n-cube.

1, 1, 2, 1, 6, 6, 1, 18, 42, 26, 1, 58, 252, 344, 150, 1, 190, 1420, 3380, 3230, 1082, 1, 614, 7770, 29200, 47130, 34452, 9366

Offset: 0

Tilman Piesk, Mar 11 2018

n-place formulas in first-order logic like Ax Ey P(x, y) can be ordered by implication. This Hasse diagram can be interpreted as an n-dimensional convex polytope with face dimensions ranging from 0 (the vertices) to n (the polytope itself).
The right diagonal (n-k = 0, number of vertices) is A000629, which is twice an ordered Bell number (A000670) for n>0.
The second right diagonal (n-k = 1, number of edges) is A300693.
The second left diagonal (k = 1, number of facets) is 2, 6, 18, 58, 190, 614, ... (not to be confused with A151282 or A193777).
The third left diagonal (k = 2, number of ridges) is 6, 42, 252, 1420, 7770, ...
The row sums are A300701. The central diagonal starts 1, 6, 252, 29200 and the row maxima start 1, 2, 6, 42, 344, 3380, 47130.
The corresponding triangle for hypercubes is A013609, and its row sums are A000244 (powers of 3). That for permutohedra is A019538, and its row sums are A000670 (ordered Bell numbers).

			First rows of the triangle:
  k      0     1     2     3     4     5     6         sums = A300701
n
0        1                                                1
1        1     2                                          3
2        1     6     6                                   13
3        1    18    42    26                             87
4        1    58   252   344   150                      805
5        1   190  1420  3380  3230  1082               9303
6        1   614  7770 29200 47130 34452  9366       128533
T(3, 3-1) = T(3, 2) = 42 is the number of 1-faces (edges) of the concertina 3-cube. It has 26 vertices, 42 edges, 18 faces and 1 cell.
In the reflected triangle the column number is the dimension of the counted faces:
  n-k    0     1     2     3     4     5     6
n
0        1
1        2     1
2        6     6     1
3       26    42    18     1
4      150   344   252    58     1
5     1082  3230  3380  1420   190     1
6     9366 34452 47130 29200  7770   614     1

Tilman Piesk, Formulas in predicate logic (Wikiversity)
Tilman Piesk, Skeleton and solid representation of a concertina cube
Tilman Piesk, SAGE code used to generate the sequence

Cf. A300701, A000629, A300693.
Cf. A013609, A000244 (for hypercubes).
Cf. A019538, A000670 (for permutohedra).

A300694 a(n) = number of edges in a cocoon concertina n-cube.

Original entry on oeis.org

0, 1, 13, 139, 1605, 20741

Offset: 0

Tilman Piesk, Mar 24 2018

n-place formulas in first-order logic like Ax Ey P(x, y) or Ex P(x, x) can be ordered by implication. This Hasse diagram has A300696(n) vertices and a(n) edges.

The corresponding sequence for convex concertina n-cubes is A300693.

			The cocoon concertina square has the A300693(2) = 6 outer and 7 inner edges, giving a(n) = 13 in total.

Tilman Piesk, Formulas in predicate logic (Wikiversity)
Tilman Piesk, Image of a cocoon concertina square with 13 edges
Tilman Piesk, Lists of edges for n=2..5
Tilman Piesk, Python code used to generate the sequence

Cf. A300696, A300693.

cp's OEIS Frontend

A300700 Triangle read by rows: T(n, n-k) = number of k-faces of the concertina n-cube.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs