Stirling Numbers of the Second Kind and Nonattacking Rooks

The number of possible configurations of nonattacking rooks on a triangular chessboard can be counted by the Stirling numbers of the second kind . In particular, for rooks on a board with side length , the number of configurations is .

THINGS TO TRY

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Snapshot 1: there is only one configuration of nonattacking rooks on a triangle board with side —along the diagonal—which corresponds to the relation
Snapshot 2: a single rook on a board with side can be placed on any of the squares, so that
Snapshot 3: valid configurations can be derived recursively: the configurations for rooks on a board with side can be constructed from a combination of rooks on a board with side (adding an empty column to each existing configuration) and rooks on a board with side (adding a column with one rook in each valid square), which corresponds to the recurrence relation
Reference
[1] M. B. Wells, Elements of Combinatorial Computing, Oxford: Pergamon Press, 1971 p. 185.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.