%I #138 Mar 31 2023 15:29:21

%S 1,2,12,184,8512,1262816,575780564,789360053252,3266598486981642,

%T 41044208702632496804,1568758030464750013214100,

%U 182413291514248049241470885236,64528039343270018963357185158482118,69450664761521361664274701548907358996488

%N Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of an n X n grid.

%C The length of the path varies.

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 5.10, pp. 331-338.

%D Guttmann A J and Jensen I 2022 Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices Journal of Physics A: Mathematical and Theoretical 55 012345, (33pp) ; arXiv:2208.06744, Aug 2022.

%D D. E. Knuth, 'Things A Computer Scientist Rarely Talks About,' CSLI Publications, Stanford, CA, 2001, pages 27-28.

%D D. E. Knuth, The Art of Computer Programming, Section 7.1.4.

%D Shin-ichi Minato, The power of enumeration - BDD/ZDD-based algorithms for tackling combinatorial explosion, Chapter 3 of Applications of Zero-Suppressed Decision Diagrams, ed. T. Satsoa and J. T. Butler, Morgan & Claypool Publishers, 2014

%D Shin-ichi Minato, Counting by ZDD, Encyclopedia of Algorithms, 2014, pp. 1-6.

%D Netnews group rec.puzzles, Frequently Asked Questions (FAQ) file. (Science Section).

%H I. Jensen, H. Iwashita, and R. Spaans, <a href="/A007764/b007764.txt">Table of n, a(n) for n = 1..27</a> (I. Jensen computed terms 1 to 20, H. Iwashita computed 21 and 22, R. Spaans computed 23 to 25, and H. Iwashita computed 26 and 27)

%H M. Bousquet-Mélou, A. J. Guttmann and I. Jensen, <a href="http://arXiv.org/abs/cond-mat/0506341">Self-avoiding walks crossing a square</a>, arXiv:cond-mat/0506341 [cond-mat.stat-mech], 2005.

%H Doi, Maeda, Nagatomo, Niiyama, Sanson, Suzuki, et al., <a href="http://www.youtube.com/watch?v=Q4gTV4r0zRs">Time with class! Let's count!</a> [Youtube-animation demonstrating this sequence. In Japanese with English translation]

%H Steven R. Finch, <a href="http://web.archive.org/web/20010207195347/http://www.mathsoft.com/asolve/constant/cnntv/cnntv.html">Self-Avoiding-Walk Connective Constants</a>

%H Anthony J. Guttmann and Iwan Jensen, <a href="https://arxiv.org/abs/2211.14482">The gerrymander sequence, or A348456</a>, arXiv:2211.14482 [math.CO], 2022.

%H H. Iwashita, J. Kawahara, and S. Minato, <a href="http://www-alg.ist.hokudai.ac.jp/~thomas/TCSTR/tcstr_12_60/tcstr_12_60.pdf">ZDD-Based Computation of the Number of Paths in a Graph</a>, TCS Technical Report, TCS-TR-A-12-60, Hokkaido University, September 18, 2012.

%H H. Iwashita, Y. Nakazawa, J. Kawahara, T. Uno, and S. Minato, <a href="http://www-alg.ist.hokudai.ac.jp/~thomas/TCSTR/tcstr_13_64/tcstr_13_64.pdf">Efficient Computation of the Number of Paths in a Grid Graph with Minimal Perfect Hash Functions</a>, TCS Technical Report, TCS-TR-A-13-64, Hokkaido University, April 26, 2013.

%H I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/saw/SAW_ser.html">Series Expansions for Self-Avoiding Walks</a>

%H Shin-ichi Minato, <a href="https://dx.doi.org/10.1587/transinf.2016LOI0002">Power of Enumeration - Recent Topics on BDD/ZDD-Based Techniques for Discrete Structure Manipulation</a>, IEICE Transactions on Information and Systems, Volume E100.D (2017), Issue 8, p. 1556-1562.

%H OEIS Wiki, <a href="/wiki/Self-avoiding_walks#Self-avoiding walks on an (n+1) X (n+1) square lattice starting from (0,0) and ending at (n,n)">Self-avoiding_walks</a>

%H Kohei Shinohara, Atsuto Seko, Takashi Horiyama, Masakazu Ishihata, Junya Honda, and Isao Tanaka, <a href="https://arxiv.org/abs/2002.12603">Derivative structure enumeration using binary decision diagram</a>, arXiv:2002.12603 [physics.comp-ph], 2020.

%H Ruben Grønning Spaans, <a href="https://github.com/stubbscroll/OEIS/blob/master/A007764-fast.c">C program</a>

%H Jens Weise, <a href="https://opendata.uni-halle.de/bitstream/1981185920/103345/1/Weise_Jens_Dissertation_2023.pdf">Evolutionary Many-Objective Optimisation for Pathfinding Problems</a>, Ph. D. Dissertation, Otto von Guercke Univ., Magdeburg (Germany, 2022), see p. 53.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Self-AvoidingWalk.html">Self-Avoiding Walk</a>

%e Suppose we start at (1,1) and end at (n,n). Let U, D, L, R denote steps that are up, down, left, right.

%e a(2) = 2: UR or RU.

%e a(3) = 12: UURR, UURDRU, UURDDRUU, URUR, URRU, URDRUU and their reflections in the x=y line.

%o (Python)

%o # Using graphillion

%o from graphillion import GraphSet

%o import graphillion.tutorial as tl

%o def A007764(n):

%o if n == 1: return 1

%o universe = tl.grid(n - 1, n - 1)

%o GraphSet.set_universe(universe)

%o start, goal = 1, n * n

%o paths = GraphSet.paths(start, goal)

%o return paths.len()

%o print([A007764(n) for n in range(1, 10)]) # _Seiichi Manyama_, Mar 21 2020

%Y Main diagonal of A064298.

%Y Cf. A064297, A271507, A001184, A000532.

%K nonn,walk,hard,nice

%O 1,2

%A _David Radcliffe_ and _Don Knuth_

%E Computed to n=12 by John Van Rosendale in 1981

%E Extended to n=13 by _Don Knuth_, Dec 07 1995

%E Extended to n=20 by _Mireille Bousquet-Mélou_, A. J. Guttmann and I. Jensen

%E Extended to n=22 using ZDD technique based on Knuth's The Art of Computer Programming (exercise 225 in 7.1.4) by H. Iwashita, J. Kawahara, and S. Minato, Sep 18 2012

%E Extended to n=25 using state space compression (with rank/unrank) and dynamic programming (based in I. Jensen) by _Ruben Grønning Spaans_, Feb 22 2013

%E Extended to n=26 by _Hiroaki Iwashita_, Apr 11 2013

%E Extended to n=27 by _Hiroaki Iwashita_, Nov 18 2013