%I M3229 #88 Dec 21 2020 12:12:17

%S 1,1,2,4,4,8,24,32,40,120,296,648,1328,3200,9912,25592,55920,143192,

%T 510696,1451296,3497344,10451824,38570704,118914992,315235872,

%U 1014824752,3963684496,13166130152,37846301904,130507967088,533318630936,1884550215976,5800121391936

%N Number of graceful permutations of length n.

%C Also the number of graceful labelings of the path graph P_n. - _Eric W. Weisstein_, Mar 31 2020

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D H. S. Wilf and N. Yoshimura, Ranking rooted trees and a graceful application, in Discrete Algorithms and Complexity (Proceedings of the Japan-US joint seminar, 1986, Kyoto, Japan), edited by D. Johnson, T. Nishizeki, A. Nozaki and H. S. Wilf, Academic Press, NY, 1987, pp. 341-350.

%H Michal Adamaszek, Don Knuth, <a href="/A006967/b006967.txt">Table of n, a(n) for n = 0..41</a>, a(41) from Don Knuth.

%H M. Adamaszek, <a href="https://arxiv.org/abs/math/0608513">Efficient enumeration of graceful permutations</a>, arXiv:math/0608513 [math.CO], 2006.

%H Gheorghe Coserea, <a href="/A006967/a006967.txt">Solutions for n=5</a>.

%H Gheorghe Coserea, <a href="/A006967/a006967_1.txt">Solutions for n=6</a>.

%H Gheorghe Coserea, <a href="/A006967/a006967.mzn.txt">MiniZinc model for generating solutions</a>.

%H Don Knuth, <a href="http://cs.stanford.edu/~knuth/programs/back-graceful-perms.w">This program finds all of the nonisomorphic graceful labelings of the path P_n</a>

%H Don Knuth, <a href="http://cs.stanford.edu/~knuth/programs/zddl-graceful-perms.w">This program outputs ZDDL for all of the nonisomorphic graceful labelings of the path P_n</a>

%H Md Masbaul Alam Polash, M. A. Hakim Newton, Abdul Sattar, <a href="https://doi.org/10.1007/s10601-016-9261-y">Constraint-directed search for all-interval series</a>, Constraints, July 2017, Volume 22, Issue 3, pp 403-431. See page 426.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GracefulLabeling.html">Graceful Labeling</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GracefulPermutation.html">Graceful Permutation</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PathGraph.html">Path Graph</a>

%H J. Wodlinger, <a href="http://summit.sfu.ca/item/12222">Costas arrays, Golomb rulers and wavelength isolation sequence pairs, M.S. Dissertation</a>, Math. Dept., Simon Fraser University, Spring 2012; - From _N. J. A. Sloane_, Oct 15 2012

%F a(n) = n! - A084894(n). - _Jon Perry_, Jun 10 2003

%o (CWEB) programs by _Don Knuth_: see links above

%Y Cf. A084894.

%K nonn,nice,hard

%O 0,3

%A _N. J. A. Sloane_

%E n=2 term corrected June 1996

%E a(11)-a(20) from Robert Aldred and _Brendan McKay_

%E More terms from Michal Adamaszek (aszek(AT)mimuw.edu.pl), Aug 22 2006

%E a(0)=1 prepended by _Alois P. Heinz_, Jan 31 2020

%E a(41)=1032009647743958000 from _Don Knuth_, Sep 10 2020