|
| |
|
|
A151498
|
|
Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2 n steps taken from {(-1, 1), (1, -1), (1, 0)}.
|
|
0
|
|
|
|
1, 1, 3, 12, 57, 301, 1707, 10191, 63244, 404503, 2650293, 17709684, 120288313, 828352036, 5771747783, 40625485570, 288482116987, 2064429518054, 14874855533504, 107832596542894, 785986247826371, 5757192302807027, 42357833323697589, 312901369167191854, 2319946973815289676
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
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.
|
|
|
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
|
|
|
|
STATUS
|
approved
|
| |
|
|
