A338986 -id:A338986 - cp's OEIS frontend

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

Don Knuth, Dec 20 2020

nonn

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.

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

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

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

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.

cp's OEIS Frontend

A338988 Number of rooted graceful labelings of the path P_n.

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