A338987 Number of rooted graceful labelings with n edges.

1, 1, 2, 6, 24, 108, 596, 3674, 26068, 202470, 1753884, 16435754, 168174596, 1842418704, 21757407158, 272771715272, 3649515044178, 51532670206504, 770442883634326, 12093451621846094, 199856952123506304, 3452120352032161404, 62471981051497913826, 1177664861561125869100, 23163177237781937250558

Offset: 0

Author

Don Knuth, Dec 20 2020

Keywords

• nonn

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)|

To generate such a labeling, choose one of the k+1 edges 0--(n-k), 1--(n+1-k), ..., k--(n-k), for each k=0, 1, ..., n-1, provided that at least one of the endpoints has already appeared in a previously chosen edge when k>0.

Examples

			a(5) = 108 < 120 = 5!, because 0--5,0--4,0--3,3--5,1--2 and 0--5,1--5,2--5,0--2,1--3 are forbidden, as well as five each beginning with 0--5,0--4,2--5,1--3 and 0--5,1--4,0--3,2--4.
		

Links

• David A. Sheppard, The factorial representation of major balanced labelled graphs, Discrete Math., Vol. 15, No. 4 (1976), 379-388.

Crossrefs

Without rootedness, the number of graceful labelings is n!, A000142, as observed by Sheppard in 1976.

Programs

Python

def solve(d, m_in):
    global _n, _cache
    args = (d, m_in)
    if args in _cache:
        return _cache[args]
    if d == 0:
        rv = 1
    else:
        rv = 0
        m_test = 1 | (1<Bert Dobbelaere, Dec 23 2020

Extensions

a(17)-a(24) from Bert Dobbelaere, Dec 23 2020

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