D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of n-ominoes, Canad. J. Math. 25 (1973) 585-602.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....if you counted them too long. Indeed, enumerating animals on a periodic infinite graph seems to be a nightmarish problem. To our knowledge, Partially supported by EC grant CHRX CT93 0400 and PRC Math ematiques et Informatique . the most precise rigorous result in this field is the following [18]: there exists a constant K such that, if an denotes the number of square lattice animals having n cells, then a 1=n n tends to K when n tends to infinity. Finding lower and upper bounds for K is difficult, and the first digit of K is not yet known: 3; 87 K 4; 65: Some non rigorous but ....

D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of n-ominoes, Canad. J. Math. 13 (1973) 585--602.

....over the last 40 years, these problems are completely open. Though, a few asymptotic results are known. For example, let p(n) denote the number of polyominoes having n cells. Klarner and Rivest proved that (p(n) 1=n tends to a limit K, which satisfies the following inequality: 3; 87 K 4; 65 [17]. The situation could hardly be worse, since the first digit of K is not even known. The difficulty of these questions has led to the study of various restricted classes of polyominoes or polygons. Most of them can be defined by combining two notions: a geometric notion of convexity, and a ....

D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of n-ominoes, Canad. J. Math. 25 No. 3 (1973) 585-602.

....of n cells on the square lattice. A concatenation argument [14] shows that there exists a constant , called the connective constant , such that lim n 1 (c n ) 1=n = The exact value of is unknown, though numerical studies [6] have shown that 4:06. The best published 1 bounds on [16] are 3:72 4:65: It is a measure of the complexity of the problem that not even the rst digit of is known rigorously. Given the diOEculty in solving this problem, what rigorous work can be done towards better understanding polyominoes Perhaps the most fruitful work has been in the ....

D. A. Klarner and R. L. Rivest. A procedure for improving the upper bound for the number of n-ominoes. Can. J. Math., 25:585602, 1973.

....and PRC Math ematiques et Informatique . An extended version of this paper, including proofs, will appear later on. Figure 1: An animal on the square lattice. Enumerating these animals seems to be a very difficult problem. To our knowledge, the most precise result is the following [14]: there exists a constant K such that, if a n denotes the number of square lattice animals having n vertices, then a 1=n n tends to K when n tends to infinity. Finding lower and upper bounds for K is difficult, and not even the first digit of K is known: 3; 87 K 4; 65: The enumeration of ....

D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of n-ominoes, Canad. J. Math. 13 (1973) 585--602.

....over the last 40 years, these problems are completely open. Though, a few asymptotic results are known. For example, let p(n) denote the number of polyominoes having n cells. Klarner and Rivest proved that (p(n) 1=n tends to a limit K, which satisfies the following inequality: 3; 87 K 4; 65 [17]. The situation could hardly be worse, since the first digit of K is not even known. The difficulty of these questions has led to the study of various restricted classes of polyominoes or polygons. Most of them can be defined by combining two notions: a geometric notion of convexity, and a ....

D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of n-ominoes, Canad. J. Math. 25 No. 3 (1973) 585-602.

....if you counted them too long. Indeed, enumerating animals on a periodic infinite graph seems to be a nightmarish problem. To our knowledge, Partially supported by EC grant CHRX CT93 0400 and PRC Math ematiques et Informatique . the most precise rigorous result in this field is the following [18]: there exists a constant K such that, if an denotes the number of square lattice animals having n cells, then a 1=n n tends to K when n tends to infinity. Finding lower and upper bounds for K is difficult, and the first digit of K is not yet known: 3; 87 K 4; 65: Some non rigorous but ....

D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of n-ominoes, Canad. J. Math. 13 (1973) 585--602.

....over the last 40 years, these problems are completely open. Though, a few asymptotic results are known. For example, let p(n) denote the number of polyominoes having n cells. Klarner and Rivest proved that (p(n) 1=n tends to a limit K, which satisfies the following inequality: 3; 87 K 4; 65 [16]. The situation could hardly be worse, since the first digit of K is not even known. The difficulty of these questions has led to the study of various restricted classes of polyominoes or polygons. Most of them can be defined by combining two notions: a geometric notion of convexity, and a ....

D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of n-ominoes, Canad. J. Math. 25 No. 3 (1973) 585-602.

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D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of n-ominoes, Canad. J. Math. 25 (1973) 585-602.

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