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A300700 Triangle read by rows: T(n, n-k) = number of k-faces of the concertina n-cube. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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).
LINKS
Tilman Piesk, Formulas in predicate logic (Wikiversity)
Tilman Piesk, Skeleton and solid representation of a concertina cube
EXAMPLE
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
CROSSREFS
Cf. A013609, A000244 (for hypercubes).
Cf. A019538, A000670 (for permutohedra).
Sequence in context: A019538 A269646 A269336 * A046521 A104684 A060538
KEYWORD
nonn,tabl,more
AUTHOR
Tilman Piesk, Mar 11 2018
STATUS
approved

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Last modified March 28 08:12 EDT 2024. Contains 371236 sequences. (Running on oeis4.)