

A006967


Number of graceful permutations of length n.
(Formerly M3229)


10



1, 1, 2, 4, 4, 8, 24, 32, 40, 120, 296, 648, 1328, 3200, 9912, 25592, 55920, 143192, 510696, 1451296, 3497344, 10451824, 38570704, 118914992, 315235872, 1014824752, 3963684496, 13166130152, 37846301904, 130507967088, 533318630936, 1884550215976, 5800121391936
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OFFSET

0,3


COMMENTS

Also the number of graceful labelings of the path graph P_n.  Eric W. Weisstein, Mar 31 2020


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. S. Wilf and N. Yoshimura, Ranking rooted trees and a graceful application, in Discrete Algorithms and Complexity (Proceedings of the JapanUS joint seminar, 1986, Kyoto, Japan), edited by D. Johnson, T. Nishizeki, A. Nozaki and H. S. Wilf, Academic Press, NY, 1987, pp. 341350.


LINKS

Michal Adamaszek, Don Knuth, Table of n, a(n) for n = 0..41, a(41) from Don Knuth.
M. Adamaszek, Efficient enumeration of graceful permutations, arXiv:math/0608513 [math.CO], 2006.
Gheorghe Coserea, Solutions for n=5.
Gheorghe Coserea, Solutions for n=6.
Gheorghe Coserea, MiniZinc model for generating solutions.
Don Knuth, This program finds all of the nonisomorphic graceful labelings of the path P_n
Don Knuth, This program outputs ZDDL for all of the nonisomorphic graceful labelings of the path P_n
Md Masbaul Alam Polash, M. A. Hakim Newton, Abdul Sattar, Constraintdirected search for allinterval series, Constraints, July 2017, Volume 22, Issue 3, pp 403431. See page 426.
Eric Weisstein's World of Mathematics, Graceful Labeling
Eric Weisstein's World of Mathematics, Graceful Permutation
Eric Weisstein's World of Mathematics, Path Graph
J. Wodlinger, Costas arrays, Golomb rulers and wavelength isolation sequence pairs, M.S. Dissertation, Math. Dept., Simon Fraser University, Spring 2012;  From N. J. A. Sloane, Oct 15 2012


FORMULA

a(n) = n!  A084894(n).  Jon Perry, Jun 10 2003


PROG

(CWEB) programs by Don Knuth: see links above


CROSSREFS

Cf. A084894.
Sequence in context: A080007 A239649 A264190 * A296229 A322175 A298117
Adjacent sequences: A006964 A006965 A006966 * A006968 A006969 A006970


KEYWORD

nonn,nice,hard


AUTHOR

N. J. A. Sloane


EXTENSIONS

n=2 term corrected June 1996
a(11)a(20) from Robert Aldred and Brendan McKay
More terms from Michal Adamaszek (aszek(AT)mimuw.edu.pl), Aug 22 2006
a(0)=1 prepended by Alois P. Heinz, Jan 31 2020
a(41)=1032009647743958000 from Don Knuth, Sep 10 2020


STATUS

approved



