Séminaire Lotharingien de Combinatoire, 82B.22 (2019), 8 pp.
Arthur L. Gershon
On the asymptotic enumeration of restricted strip arrangements of a chessboard
Abstract.
We are concerned with the number T(m,n) of ways to arrange 1 x k non-overlapping strips on an m x n chessboard (not necessarily fully covering the chessboard) such that there is at most one strip with its longest side horizontal in each row and at most one vertical strip in each row. While generating functions for T(m,n) have been computed in some cases using the transfer matrix method, a general formula has proved elusive; in lieu of this, we provide a way to estimate T(m,n) asymptotically. In the case where one dimension m of the chessboard is fixed and the other dimension n tends towards infinity, we are able to determine an asymptotically equivalent product formula for T(m,n) as n -> infinity.
Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.
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