A332389 - OEIS

%I #21 Feb 11 2020 12:05:27

%S 1,1,2,1,4,3,1,8,7,4,1,16,18,10,5,1,32,47,28,13,6,1,64,123,82,38,16,7,

%T 1,128,322,244,117,48,19,8,1,256,843,730,370,152,58,22,9,1,512,2207,

%U 2188,1186,496,187,68,25,10,1,1024,5778,6562,3827,1648,622,222,78,28,11

%N Number A(n,w) of circular Dyck paths with n entries, and width at most w.

%C A(n,w) is the number of circular Dyck paths of size n, and width at most w.

%C This is also the number of circular area lists, a_1, a_2, ..., a_n such that 0 <= a_i <= w-1, and a_{i+1} < a_i + 1, for all 1 <= i <= n, and the index i is taken modulo n.

%C The values of w are given by the row index.

%C A(n,w) is given by summing binomial(2*n - 1, n - 1 - (w+2) k) - binomial(2*n - 1, n + j + (w+2)*k) over k=1..w and k over all integers.

%H Per Alexandersson, Svante Linusson and Samu Potka, <a href="https://arxiv.org/abs/1903.01327">The cyclic sieving phenomenon on circular Dyck paths</a>, arXiv:1903.01327 [math.CO], 2019.

%H Per Alexandersson, Svante Linusson and Samu Potka, <a href="https://doi.org/10.37236/8720">The cyclic sieving phenomenon on circular Dyck paths</a>, Electronic Journal of Combinatorics 26, No.4 (2019).

%F A(n,w) = Sum_{k=-2*(n+2)..2*(n+2)} Sum_{j=1..w} binomial(2n-1, n-1-(w+2)*k) - binomial(2*n-1, n + j + (w+2)*k).

%e The table begins as

%e 1,    2,    3,    4,    5, ...

%e 1,    4,    7,    10,   13, ...

%e 1,    8,    18,   28,   38, ...

%e 1,    16,   47,   82,   117, ...

%e 1,    32,   123,  244,  370, ...

%e ...

%e A(5,3)=123 and a few of the corresponding circular area lists are 00000, 10000,...,12210,...,12222, 22222.

%t CircularDyckPaths[n_, w_] := With[{d = w + 2},

%t    Sum[Binomial[2 n - 1, n - 1 - d s] -

%t      Binomial[2 n - 1, n + j + d s]

%t     , {j, w},

%t     {s, -2 (n + 2), 2 (n + 2)}]

%t    ];

%t Table[

%t CircularDyckPaths[n, w]

%t , {n, 1, 10}, {w, 1, 10}]

%Y A194460 is the diagonal.

%K nonn,tabl

%O 1,3

%A _Per W. Alexandersson_, Feb 10 2020