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A224993 Non-crossing, non-nesting, 4-colored permutations on {1,2,...,n}.

%I A224993 #32 Jan 31 2024 10:56:31
%S A224993 1,4,32,352,4736,72832,1226240,21948928,409192448,7833143296,
%T A224993 152494727168,3000118779904,59406517698560,1180988766453760,
%U A224993 23534128521936896,469655122210324480,9380774946206646272,187467580232576794624,3747576648059504820224
%N A224993 Non-crossing, non-nesting, 4-colored permutations on {1,2,...,n}.
%C A224993 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.
%H A224993 Lily Yen, <a href="/A224993/b224993.txt">Table of n, a(n) for n = 0..99</a>
%H A224993 Lily Yen, <a href="http://arxiv.org/abs/1211.3472">Crossings and Nestings for Arc-Coloured Permutations</a> and <a href="https://doi.org/10.46298/dmtcs.2339">Arc-coloured permutations</a>, PSAC 2013, Paris, France, June 24-28, Proc. DMTCS (2013) 743-754.
%H A224993 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (40,-508,2304,-2880).
%F A224993 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)).
%F A224993 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]
%e A224993 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.
%t A224993 Join[{1}, LinearRecurrence[{40, -508, 2304, -2880}, {4, 32, 352, 4736}, 20]] (* _Jean-François Alcover_, Jul 22 2018 *)
%o A224993 (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
%K A224993 nonn,easy
%O A224993 0,2
%A A224993 _Lily Yen_, Apr 24 2013