A342381 Triangle read by rows: T(n,k) is the number of symmetries of the n-dimensional hypercube that fix exactly 2*k facets; n,k >= 0.

1, 1, 1, 5, 2, 1, 29, 15, 3, 1, 233, 116, 30, 4, 1, 2329, 1165, 290, 50, 5, 1, 27949, 13974, 3495, 580, 75, 6, 1, 391285, 195643, 48909, 8155, 1015, 105, 7, 1, 6260561, 3130280, 782572, 130424, 16310, 1624, 140, 8, 1, 112690097, 56345049, 14086260, 2347716, 293454, 29358, 2436, 180, 9, 1

Offset: 0

Author

Peter Kagey, Mar 09 2021

Keywords

• nonn
• tabl

Comments

Equivalently the number of symmetries of the n-dimensional cross-polytope that fix exactly 2*k vertices.

If a facet of the hypercube is fixed, then the opposite facet must also be fixed.

Examples

			Table begins:
n\k |         0        1        2       3      4     5    6   7 8 9
----+--------------------------------------------------------------
  0 |         1
  1 |         1        1
  2 |         5        2        1
  3 |        29       15        3       1
  4 |       233      116       30       4      1
  5 |      2329     1165      290      50      5     1
  6 |     27949    13974     3495     580     75     6    1
  7 |    391285   195643    48909    8155   1015   105    7   1
  8 |   6260561  3130280   782572  130424  16310  1624  140   8 1
  9 | 112690097 56345049 14086260 2347716 293454 29358 2436 180 9 1
For the cube in n=2 dimensions (the square) there is
T(2,2) = 1 symmetry that fixes all 2*2 = 4 sides, namely the identity:
     2
   +---+
  3|   |1;
   +---+
     4
T(2,1) = 2 symmetries that fix 2*1 = 2 sides, namely horizonal/vertical flips:
     4           2
   +---+       +---+
  3|   |1 and 1|   |3;
   +---+       +---+
     2           4
and T(2,0) = 5 symmetries that fix 2*0 = 0 sides, namely rotations or diagonal flips:
     1         4         3         3            1
   +---+     +---+     +---+     +---+        +---+
  2|   |4,  1|   |3,  4|   |2,  2|   |4, and 4|   |2.
   +---+     +---+     +---+     +---+        +---+
     3         2         1         1            3
		

Links

• Peter Kagey, Rows n = 0..100, flattened
• Wikipedia, Cross-polytope
• Wikipedia, Hypercube
• Wikipedia, Hyperoctahedral group

Crossrefs

Columns and diagonals: A000354 (k=0), A161937 (k=1), A028895 (n=k+2).

Row sums are A000165.

Cf. A001147, A114320.

Programs

PARI

f(n) = sum(k=0, n, (-1)^(n+k)*binomial(n, k)*k!*2^k); \\ A000354
T(n, k) = f(n-k)*binomial(n, k); \\ Michel Marcus, Mar 10 2021

Formula

T(n,k) = A114320(2n,k)/A001147(n).

T(n,k) = A000354(n-k)*binomial(n,k).