|
|
A305624
|
|
Number of chiral pairs of rows of n colors with exactly 4 different colors.
|
|
2
|
|
|
0, 0, 0, 12, 120, 780, 4188, 20400, 93120, 409140, 1748220, 7337232, 30386160, 124696740, 508250988, 2061566400, 8331954240, 33585590580, 135115594140, 542784981552, 2178107091600, 8733341736900, 34996103558988, 140172672276000, 561255446475360, 2246716252964820, 8991948337723260, 35983044114659472, 143977928423467440, 576048972752188260, 2304607666801990188, 9219666007300387200, 36882370043723748480
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,4
|
|
COMMENTS
|
If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=4 colors used and where S2(n,k) is the Stirling subset number A008277.
G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=4 colors used.
|
|
EXAMPLE
|
For a(4) = 12, the chiral pairs are the 4! = 24 permutations of ABCD, each paired with its reverse.
|
|
MATHEMATICA
|
k=4; Table[(k!/2) (StirlingS2[n, k] - StirlingS2[Ceiling[n/2], k]), {n, 1, 40}]
|
|
PROG
|
(PARI) a(n) = my(k=4); (k!/2) * (stirling(n, k, 2) - stirling(ceil(n/2), k, 2)); \\ Michel Marcus, Jun 07 2018
|
|
CROSSREFS
|
A056455(n) is number of achiral rows of n colors with exactly 4 different colors.
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|