login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Triangle read by rows: T(n,k) is the number of grand Dyck paths of semilength n having degree of symmetry k (n >= 0, 0 <= k <= n).
0

%I #31 Feb 12 2021 18:43:43

%S 1,0,2,2,0,4,4,8,0,8,14,16,24,0,16,44,64,48,64,0,32,148,208,216,128,

%T 160,0,64,504,736,720,640,320,384,0,128,1750,2592,2672,2176,1760,768,

%U 896,0,256,6156,9280,9696,8448,6080,4608,1792,2048,0,512

%N Triangle read by rows: T(n,k) is the number of grand Dyck paths of semilength n having degree of symmetry k (n >= 0, 0 <= k <= n).

%C The degree of symmetry of a grand Dyck path is defined as the number of steps in the first half that are mirror images of steps in the second half, with respect to the reflection along a vertical line through the midpoint of the path.

%H Sergi Elizalde, <a href="https://arxiv.org/abs/2002.12874">The degree of symmetry of lattice paths</a>, arXiv:2002.12874 [math.CO], 2020.

%H Sergi Elizalde, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2020/26.html">Measuring symmetry in lattice paths and partitions</a>, Sem. Lothar. Combin. 84B.26, 12 pp (2020).

%F G.f.: 1/(2(1-u)z+sqrt(1-4z)).

%e For n=3 there are 4 grand Dyck paths with degree of symmetry equal to 0, namely uddduu, uudddu, duuudd, dduuud.

%e The triangle begins:

%e 1

%e 0 2

%e 2 0 4

%e 4 8 0 8

%e 14 16 24 0 16

%e 44 64 48 64 0 32

%e 148 208 216 128 160 0 64

%e 504 736 720 640 320 384 0 128

%Y Cf. A000079 (diagonal), A000984 (row sums).

%K nonn,tabl

%O 0,3

%A _Sergi Elizalde_, Feb 12 2021