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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.

%I A342381 #24 Jun 02 2025 15:24:10
%S A342381 1,1,1,5,2,1,29,15,3,1,233,116,30,4,1,2329,1165,290,50,5,1,27949,
%T A342381 13974,3495,580,75,6,1,391285,195643,48909,8155,1015,105,7,1,6260561,
%U A342381 3130280,782572,130424,16310,1624,140,8,1,112690097,56345049,14086260,2347716,293454,29358,2436,180,9,1
%N 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.
%C A342381 Equivalently the number of symmetries of the n-dimensional cross-polytope that fix exactly 2*k vertices.
%C A342381 If a facet of the hypercube is fixed, then the opposite facet must also be fixed.
%H A342381 Peter Kagey, <a href="/A342381/b342381.txt">Rows n = 0..100, flattened</a>
%H A342381 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cross-polytope">Cross-polytope</a>
%H A342381 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypercube">Hypercube</a>
%H A342381 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hyperoctahedral_group">Hyperoctahedral group</a>
%F A342381 T(n,k) = A114320(2n,k)/A001147(n).
%F A342381 T(n,k) = A000354(n-k)*binomial(n,k).
%e A342381 Table begins:
%e A342381 n\k |         0        1        2       3      4     5    6   7 8 9
%e A342381 ----+--------------------------------------------------------------
%e A342381   0 |         1
%e A342381   1 |         1        1
%e A342381   2 |         5        2        1
%e A342381   3 |        29       15        3       1
%e A342381   4 |       233      116       30       4      1
%e A342381   5 |      2329     1165      290      50      5     1
%e A342381   6 |     27949    13974     3495     580     75     6    1
%e A342381   7 |    391285   195643    48909    8155   1015   105    7   1
%e A342381   8 |   6260561  3130280   782572  130424  16310  1624  140   8 1
%e A342381   9 | 112690097 56345049 14086260 2347716 293454 29358 2436 180 9 1
%e A342381 For the cube in n=2 dimensions (the square) there is
%e A342381 T(2,2) = 1 symmetry that fixes all 2*2 = 4 sides, namely the identity:
%e A342381      2
%e A342381    +---+
%e A342381   3|   |1;
%e A342381    +---+
%e A342381      4
%e A342381 T(2,1) = 2 symmetries that fix 2*1 = 2 sides, namely horizonal/vertical flips:
%e A342381      4           2
%e A342381    +---+       +---+
%e A342381   3|   |1 and 1|   |3;
%e A342381    +---+       +---+
%e A342381      2           4
%e A342381 and T(2,0) = 5 symmetries that fix 2*0 = 0 sides, namely rotations or diagonal flips:
%e A342381      1         4         3         3            1
%e A342381    +---+     +---+     +---+     +---+        +---+
%e A342381   2|   |4,  1|   |3,  4|   |2,  2|   |4, and 4|   |2.
%e A342381    +---+     +---+     +---+     +---+        +---+
%e A342381      3         2         1         1            3
%o A342381 (PARI) f(n) = sum(k=0, n, (-1)^(n+k)*binomial(n, k)*k!*2^k); \\ A000354
%o A342381 T(n, k) = f(n-k)*binomial(n, k); \\ _Michel Marcus_, Mar 10 2021
%Y A342381 Columns and diagonals: A000354 (k=0), A161937 (k=1), A028895 (n=k+2).
%Y A342381 Row sums are A000165.
%Y A342381 Cf. A001147, A114320.
%K A342381 nonn,tabl
%O A342381 0,4
%A A342381 _Peter Kagey_, Mar 09 2021