A338988 Number of rooted graceful labelings of the path P_n.

1, 1, 1, 1, 1, 2, 3, 1, 3, 3, 4, 5, 7, 3, 3, 15, 4, 5, 10, 12, 12, 13, 11, 19, 13, 18, 20, 17, 15, 34, 37, 38, 18, 54, 29, 35, 29, 46, 102, 44, 101, 72, 103, 110, 89, 96, 116, 64, 176, 111, 203, 140, 87, 226, 200, 272, 157, 179, 217, 240, 247, 224, 224, 467

Offset

0,6

Comments

A graceful labeling of a graph with n edges assigns distinct labels l(v) to the vertices such that 0 <= l(v) <= n and the n differences |l(u) - l(v)| between labels of adjacent vertices are distinct. It is rooted if, in addition, either |l(u) - l(w)| > |l(u) - l(v)| for some neighbor of u or |l(v) - l(w)| > |l(u) -l(v)| for some neighbor of v, whenever |l(u) - l(v)| < n.

Two labelings of the path P_n are considered to be the same if we get one from the other by reflecting the path left<->right and/or by complementing the labels with respect to n-1. Thus we can assume that the labeling contains the subsequence (n-2)0(n-1) when n > 2.

If x[1]...x[t] is the subsequence containing all differences > k > 1, the subsequence for k-1 must be either (x[1]\pm k)x[1]...x[t] or x[1]...x[t](x[t]\pm k).

So a[n] <= 4^(n-3) for n > 2.

References

D. E. Knuth, The Art of Computer Programming, Section 7.2.2.3 (in preparation)

Links

Don Knuth, Table of n, a(n) for n = 0..173

Example

For n = 6 the a(6) = 3 canonical labelings are 140532, 214053, 231405.

Crossrefs

A338987 counts rooted graceful labelings of all graphs (times 2 when n > 1 because it also counts complementary labelings).

A338986(n) = 4*a(n) when n > 2, because it also counts complementary labelings and reverses of labelings.

Sequence in context: A200181 A121062 A045831 * A046821 A239691 A265496

Adjacent sequences: A338985 A338986 A338987 * A338989 A338990 A338991

Keyword

nonn

Author

Don Knuth, Dec 20 2020

Status

approved