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%I #26 Apr 03 2018 10:15:12
%S 1,1,2,1,6,6,1,18,42,26,1,58,252,344,150,1,190,1420,3380,3230,1082,1,
%T 614,7770,29200,47130,34452,9366
%N Triangle read by rows: T(n, n-k) = number of k-faces of the concertina n-cube.
%C 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).
%C The right diagonal (n-k = 0, number of vertices) is A000629, which is twice an ordered Bell number (A000670) for n>0.
%C The second right diagonal (n-k = 1, number of edges) is A300693.
%C The second left diagonal (k = 1, number of facets) is 2, 6, 18, 58, 190, 614, ... (not to be confused with A151282 or A193777).
%C The third left diagonal (k = 2, number of ridges) is 6, 42, 252, 1420, 7770, ...
%C 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.
%C 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).
%H Tilman Piesk, <a href="https://en.wikiversity.org/wiki/Formulas_in_predicate_logic">Formulas in predicate logic</a> (Wikiversity)
%H Tilman Piesk, <a href="https://commons.wikimedia.org/wiki/File:Concertina_cube_Hasse_diagram.png">Skeleton</a> and <a href="https://commons.wikimedia.org/wiki/File:Concertina_cube_with_direction_colors;_ortho_rhomb.png">solid</a> representation of a concertina cube
%H Tilman Piesk, <a href="https://github.com/watchduck/concertina_hypercubes/blob/master/faces.sage">SAGE code used to generate the sequence</a>
%e First rows of the triangle:
%e k 0 1 2 3 4 5 6 sums = A300701
%e n
%e 0 1 1
%e 1 1 2 3
%e 2 1 6 6 13
%e 3 1 18 42 26 87
%e 4 1 58 252 344 150 805
%e 5 1 190 1420 3380 3230 1082 9303
%e 6 1 614 7770 29200 47130 34452 9366 128533
%e 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.
%e In the reflected triangle the column number is the dimension of the counted faces:
%e n-k 0 1 2 3 4 5 6
%e n
%e 0 1
%e 1 2 1
%e 2 6 6 1
%e 3 26 42 18 1
%e 4 150 344 252 58 1
%e 5 1082 3230 3380 1420 190 1
%e 6 9366 34452 47130 29200 7770 614 1
%Y Cf. A300701, A000629, A300693.
%Y Cf. A013609, A000244 (for hypercubes).
%Y Cf. A019538, A000670 (for permutohedra).
%K nonn,tabl,more
%O 0,3
%A _Tilman Piesk_, Mar 11 2018
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