login
A224993
Non-crossing, non-nesting, 4-colored permutations on {1,2,...,n}.
1
1, 4, 32, 352, 4736, 72832, 1226240, 21948928, 409192448, 7833143296, 152494727168, 3000118779904, 59406517698560, 1180988766453760, 23534128521936896, 469655122210324480, 9380774946206646272, 187467580232576794624, 3747576648059504820224
OFFSET
0,2
COMMENTS
A225029-A225033 are sequences counting non-crossing, non-nesting, r-colored set partitions for r=3..7. Set partitions only have upper arcs, whereas permutations have upper and lower arcs in their annotated arc diagram representations.
LINKS
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations and Arc-coloured permutations, PSAC 2013, Paris, France, June 24-28, Proc. DMTCS (2013) 743-754.
FORMULA
G.f.: (1-36*x+380*x^2-1200*x^3+576*x^4)/((1-2*x)*(1-6*x)*(1-12*x)*(1-20*x)).
a(n) = 2^(n-1)*(20*3^n+7*6^n+10^n+28)/35 for n>0, a(0)=1. [Bruno Berselli, Apr 26 2013]
EXAMPLE
For n=3, a(3)=352, the number of ways to color arcs of a permutation on 3 elements in 4 colors so that arcs of the same color do not cross nor nest.
MATHEMATICA
Join[{1}, LinearRecurrence[{40, -508, 2304, -2880}, {4, 32, 352, 4736}, 20]] (* Jean-François Alcover, Jul 22 2018 *)
PROG
(PARI) Vec((1-36*x+380*x^2-1200*x^3+576*x^4)/((1-2*x)*(1-6*x)*(1-12*x)*(1-20*x)) +O(x^66)) \\ Joerg Arndt, Apr 24 2013
CROSSREFS
Sequence in context: A137432 A177750 A304911 * A357407 A319220 A347021
KEYWORD
nonn,easy
AUTHOR
Lily Yen, Apr 24 2013
STATUS
approved