A007764 Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of an n X n grid.

1, 2, 12, 184, 8512, 1262816, 575780564, 789360053252, 3266598486981642, 41044208702632496804, 1568758030464750013214100, 182413291514248049241470885236, 64528039343270018963357185158482118, 69450664761521361664274701548907358996488

Offset: 1

Author

David Radcliffe and Don Knuth

Keywords

• nonn
• walk
• hard
• nice

Comments

The length of the path varies.

Examples

			Suppose we start at (1,1) and end at (n,n). Let U, D, L, R denote steps that are up, down, left, right.
a(2) = 2: UR or RU.
a(3) = 12: UURR, UURDRU, UURDDRUU, URUR, URRU, URDRUU and their reflections in the x=y line.
		

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 5.10, pp. 331-338.
• 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. E. Knuth, 'Things A Computer Scientist Rarely Talks About,' CSLI Publications, Stanford, CA, 2001, pages 27-28.
• D. E. Knuth, The Art of Computer Programming, Section 7.1.4.
• 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
• Shin-ichi Minato, Counting by ZDD, Encyclopedia of Algorithms, 2014, pp. 1-6.
• Netnews group rec.puzzles, Frequently Asked Questions (FAQ) file. (Science Section).

Links

• I. Jensen, H. Iwashita, and R. Spaans, Table of n, a(n) for n = 1..27 (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)
• M. Bousquet-Mélou, A. J. Guttmann and I. Jensen, Self-avoiding walks crossing a square, arXiv:cond-mat/0506341 [cond-mat.stat-mech], 2005.
• Doi, Maeda, Nagatomo, Niiyama, Sanson, Suzuki, et al., Time with class! Let's count! [Youtube-animation demonstrating this sequence. In Japanese with English translation]
• Steven R. Finch, Self-Avoiding-Walk Connective Constants
• Anthony J. Guttmann and Iwan Jensen, The gerrymander sequence, or A348456, arXiv:2211.14482 [math.CO], 2022.
• H. Iwashita, J. Kawahara, and S. Minato, ZDD-Based Computation of the Number of Paths in a Graph, TCS Technical Report, TCS-TR-A-12-60, Hokkaido University, September 18, 2012.
• H. Iwashita, Y. Nakazawa, J. Kawahara, T. Uno, and S. Minato, Efficient Computation of the Number of Paths in a Grid Graph with Minimal Perfect Hash Functions, TCS Technical Report, TCS-TR-A-13-64, Hokkaido University, April 26, 2013.
• I. Jensen, Series Expansions for Self-Avoiding Walks
• Shin-ichi Minato, Power of Enumeration - Recent Topics on BDD/ZDD-Based Techniques for Discrete Structure Manipulation, IEICE Transactions on Information and Systems, Volume E100.D (2017), Issue 8, p. 1556-1562.
• Shin-ichi Minato, Motivating Problems and Algorithmic Solutions, Algorithmic Foundations for Social Advancement, Springer, Singapore (2025), 17-29.
• OEIS Wiki, Self-avoiding_walks
• Kohei Shinohara, Atsuto Seko, Takashi Horiyama, Masakazu Ishihata, Junya Honda, and Isao Tanaka, Derivative structure enumeration using binary decision diagram, arXiv:2002.12603 [physics.comp-ph], 2020.
• Ruben Grønning Spaans, C program
• Jens Weise, Evolutionary Many-Objective Optimisation for Pathfinding Problems, Ph. D. Dissertation, Otto von Guercke Univ., Magdeburg (Germany, 2022), see p. 53.
• Eric Weisstein's World of Mathematics, Self-Avoiding Walk

Crossrefs

Main diagonal of A064298.

Cf. A064297, A271507, A001184, A000532.

Programs

Python

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A007764(n):
    if n == 1: return 1
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    start, goal = 1, n * n
    paths = GraphSet.paths(start, goal)
    return paths.len()
print([A007764(n) for n in range(1, 10)])  # Seiichi Manyama, Mar 21 2020

Extensions

Computed to n=12 by John Van Rosendale in 1981

Extended to n=13 by Don Knuth, Dec 07 1995

Extended to n=20 by Mireille Bousquet-Mélou, A. J. Guttmann and I. Jensen

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

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

Extended to n=26 by Hiroaki Iwashita, Apr 11 2013

Extended to n=27 by Hiroaki Iwashita, Nov 18 2013