|
%I #22 Mar 10 2021 14:06:43
%S 1,1,1,5,2,1,29,15,3,1,233,116,30,4,1,2329,1165,290,50,5,1,27949,
%T 13974,3495,580,75,6,1,391285,195643,48909,8155,1015,105,7,1,6260561,
%U 3130280,782572,130424,16310,1624,140,8,1,112690097,56345049,14086260,2347716,293454,29358,2436,180,9,1
%N 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
%C Equivalently the number of symmetries of the n-dimensional cross-polytope that fix exactly 2*k vertices.
%C If a facet of the hypercube is fixed, then the opposite facet must also be fixed.
%H Peter Kagey, <a href="/A342381/b342381.txt">Rows n = 0..100, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cross-polytope">Cross-polytope</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypercube">Hypercube</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hyperoctahedral_group">Hyperoctahedral group</a>
%F T(n,k) = A114320(2n,k)/A001147(n).
%F T(n,k) = A000354(n-k)*binomial(n,k).
%e Table begins:
%e n\k | 0 1 2 3 4 5 6 7 8 9
%e ----+--------------------------------------------------------------
%e 0 | 1
%e 1 | 1 1
%e 2 | 5 2 1
%e 3 | 29 15 3 1
%e 4 | 233 116 30 4 1
%e 5 | 2329 1165 290 50 5 1
%e 6 | 27949 13974 3495 580 75 6 1
%e 7 | 391285 195643 48909 8155 1015 105 7 1
%e 8 | 6260561 3130280 782572 130424 16310 1624 140 8 1
%e 9 | 112690097 56345049 14086260 2347716 293454 29358 2436 180 9 1
%e For the cube in n=2 dimensions (the square) there is
%e T(2,2) = 1 symmetry that fixes all 2*2 = 4 sides, namely the identity:
%e 2
%e +---+
%e 3| |1;
%e +---+
%e 4
%e T(2,1) = 2 symmetries that fix 2*1 = 2 sides, namely horizonal/vertical flips:
%e 4 2
%e +---+ +---+
%e 3| |1 and 1| |3;
%e +---+ +---+
%e 2 4
%e and T(2,0) = 5 symmetries that fix 2*0 = 0 sides, namely rotations or diagonal flips:
%e 1 4 3 3 1
%e +---+ +---+ +---+ +---+ +---+
%e 2| |4, 1| |3, 4| |2, 2| |4, and 4| |2.
%e +---+ +---+ +---+ +---+ +---+
%e 3 2 1 1 3
%o (PARI) f(n) = sum(k=0, n, (-1)^(n+k)*binomial(n, k)*k!*2^k); \\ A000354
%o T(n, k) = f(n-k)*binomial(n, k); \\ _Michel Marcus_, Mar 10 2021
%Y Columns and diagonals: A000354 (k=0), A161937 (k=1), A028895 (n=k+2).
%Y Row sums are A000165.
%Y Cf. A001147, A114320.
%K nonn,tabl
%O 0,4
%A _Peter Kagey_, Mar 09 2021
|
