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A193694 Triangle T(n,k), n>=0, 1<=k<=C(n), read by rows: T(n,k) = number of elements incomparable to the k-th path in the poset of Dyck paths of semilength n ordered by inclusion. 4
0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 4, 3, 4, 6, 4, 5, 4, 2, 2, 6, 2, 0, 0, 0, 13, 9, 16, 24, 9, 17, 13, 16, 21, 21, 20, 21, 24, 13, 20, 17, 19, 23, 16, 19, 16, 13, 14, 20, 13, 10, 11, 24, 23, 21, 14, 12, 21, 10, 4, 3, 24, 11, 3, 0, 0, 0, 41, 27, 53, 80, 24, 54, 40, 59, 80, 66, 75, 86, 99, 27 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,11
LINKS
Alois P. Heinz, Rows n = 0..9, flattened
EXAMPLE
Dyck paths of semilength n=3 listed in lexicographic order:
. /\
. /\ /\ /\/\ / \
. /\/\/\ /\/ \ / \/\ / \ / \
. 101010 101100 110010 110100 111000
. k = (1) (2) (3) (4) (5)
.
Only paths (2) and (3) are incomparable, thus row 3 = [0, 1, 1, 0, 0].
Triangle begins:
0;
0;
0, 0;
0, 1, 1, 0, 0;
0, 4, 3, 4, 6, 4, 5, 4, 2, 2, 6, 2, 0, 0;
0, 13, 9, 16, 24, 9, 17, 13, 16, 21, 21, 20, 21, 24, 13, 20, 17, 19, ...
MAPLE
d:= proc(n, l) local m; m:= nops(l);
`if`(n=m, [l], [seq(d(n, [l[], j])[],
j=`if`(m=0, 1, max(m+1, l[-1]))..n)])
end:
le:= proc(x, y) local i;
for i to nops(x) do if x[i]>y[i] then return false fi od; true
end:
T:= proc(n) option remember; local l;
l:= d(n, []);
seq(add(`if`(le(l[i], l[j]) or le(l[j], l[i]), 0, 1),
i=1..nops(l)), j=1..nops(l))
end:
seq(T(n), n=0..6);
MATHEMATICA
d[n_, l_] := d[n, l] = Module[{m}, m = Length[l]; If[n == m, {l}, Flatten[#, 1]&@Table[d[n, Append[l, j]], {j, If[m == 0, 1, Max[m + 1, Last[l]]], n}]]];
le[x_, y_] := Module[{i}, For[i = 1, i <= Length[x], i++, If[x[[i]] > y[[i]], Return[False]]]; True];
T[n_] := T[n] = Module[{l}, l = d[n, {}]; Table[Sum[ If[le[l[[i]], l[[j]]] || le[l[[j]], l[[i]]], 0, 1], {i, 1, Length[l]}], {j, 1, Length[l]}]];
Table[T[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Oct 08 2022, after Alois P. Heinz *)
CROSSREFS
Row lengths give A000108.
Sequence in context: A135103 A369052 A282822 * A354471 A117893 A354346
KEYWORD
nonn,look,tabf
AUTHOR
Alois P. Heinz, Aug 02 2011
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)