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A338987 Number of rooted graceful labelings with n edges. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

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.

LINKS

Table of n, a(n) for n=0..24.

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

EXAMPLE

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.

PROG

(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<<d)

........for p in range(_n + 1 - d):

............if m_in & m_test:

................rv += solve(d-1, m_in | m_test)

............m_test <<= 1

...._cache[args] = rv

....return rv

def a338987(n):

....global _cache, _n

...._cache, _n = {}, n

....return 1 if n<2 else 2*solve(n-2, 3|(1<<n))

# Bert Dobbelaere, Dec 23 2020

CROSSREFS

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

Sequence in context: A189840 A189255 A324591 * A174076 A230695 A177519

Adjacent sequences:  A338984 A338985 A338986 * A338988 A338989 A338990

KEYWORD

nonn

AUTHOR

Don Knuth, Dec 20 2020

EXTENSIONS

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

STATUS

approved

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Last modified January 27 10:39 EST 2022. Contains 350607 sequences. (Running on oeis4.)