OFFSET
0,3
COMMENTS
From Henri Mühle, Nov 27 2020: (Start)
a(n) is also the sum over all parabolic Catalan objects associated with parabolic quotients of the symmetric group S_n. The parabolic quotients of S_n are indexed by compositions of n. If alpha=(a_1,a_2,..., a_r) is a composition of n, consider the Dyck path v_alpha = N^{a_1}E^{a_1}N^{a_2}E^{a_2}...N^{a_r}E^{a_r}. The number of parabolic Catalan objects Cat(alpha) associated with alpha equals the number of Dyck paths of semilength n weakly above v_alpha.
For instance, if n=3, there are four compositions: alpha_1=(3), alpha_2=(2,1), alpha_3=(1,2), alpha_4=(1,1,1). Then, a(3) = Sum_{i=1..4} Cat(alpha_i) = 1+3+3+5 = 12.
(End)
LINKS
Nantel Bergeron, Cesar Ceballos, Vincent Pilaud, Hopf dreams, arXiv:1807.03044 [math.CO], 2018. See p. 19.
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
Cesar Ceballos, Wenjie Fang, Henri Mühle, The Steep-Bounce zeta map in Parabolic Cataland, arXiv:1903.08515 [math.CO], 2019. See pp. 32ff.
MATHEMATICA
aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n]]; Table[Sum[aux[0, k, 2 n], {k, 0, 2 n}], {n, 0, 25}]
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Manuel Kauers, Nov 18 2008
STATUS
approved