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A007846 There are three equivalent descriptions: 1. Number of (horizontally or vertically) connected arrays of 1..n on rectangular grid (otherwise zero) with only one local maximum. 2. Number of n-polyominoes labeled 1...n such that each successive labeled cell is the neighbor of a previously labeled cell. 3. Number of connected n-step paths on a rectangular lattice, diagonal or repeated steps not allowed. 13

%I #9 Mar 31 2012 12:34:49

%S 1,1,4,24,176,1504,14560,156768,1852512,23783264,329070176,4874845920,

%T 76898357216,1285734871520,22695759641440,421508294003424,

%U 8211642378316768

%N There are three equivalent descriptions: 1. Number of (horizontally or vertically) connected arrays of 1..n on rectangular grid (otherwise zero) with only one local maximum. 2. Number of n-polyominoes labeled 1...n such that each successive labeled cell is the neighbor of a previously labeled cell. 3. Number of connected n-step paths on a rectangular lattice, diagonal or repeated steps not allowed.

%C Description #1 and the extended sequence are from _R. H. Hardin_, Oct 25 2003. The sequence was originally submitted by Joel Yellin (yellin(AT)soe.ucsc.edu), who referenced the literature on random sequential adsorption (RSA) and used description #3.

%C Mapping between Descriptions 1 and 3: Renumber 1..n in Description #1 as n..1, so that there is only one local minimum.

%D A. Baram and D. Kutasov, "On the dynamics of random sequential absorption," J. Physics A: Math. Gen. 22, L251-L254 (1989).

%D Y. Fan and J. K. Percus, Use of Model Solutions in RSA on a Lattice, Phys. Rev. Lett. 67, 1677 (1991).

%e Examples for description #1:

%e For n=2 there are 4 possibilities:

%e .................

%e ..1..2..12..21...

%e ..2..1...........

%e .................

%e For n=3 there are 24 possibilities:

%e ....................

%e .123..132..231..321.

%e ....................

%e .1..1..2..3.........

%e .2..3..3..2.........

%e .3..2..1..1.........

%e .........................

%e .12.21..1.32.3...3.1..23.

%e ..3.3..32..1.21.12.23.1..

%e .........................

%e .31.13..1.23.2...2.1..32.

%e .2...2.23..1.31.13.32.1..

%e .........................

%e Examples for description #3. n=3: Place step #3 on each of six possible nearest neighbor (NN) sites of the 4 n=2 configurations given below for description #1. Total instances = 4 X 6 = 24. n=4: Place step #4 on each of 8 NN sites of 8 (straight line) configurations of the form 123. Place step #4 on each of 7 NN sites of 16 configurations of the form 12/3 (with the 3 under the 1). Total instances = 8 X 8 + 7 X 16 = 176.

%Y If the numbers are constrained to be on a line: A000079, in a square array: A087518, in an n X k array: A087783.

%K nonn,nice

%O 0,3

%A Joel Yellin (yellin(AT)soe.ucsc.edu)

%E New description and more terms from _R. H. Hardin_, Oct 25 2003

%E Entry revised following comments from Joel Yellin (yellin(AT)soe.ucsc.edu), Mar 04, 2004

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