cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

1, 1, 4, 24, 176, 1504, 14560, 156768, 1852512, 23783264, 329070176, 4874845920, 76898357216, 1285734871520, 22695759641440, 421508294003424, 8211642378316768
Offset: 0

Views

Author

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

Keywords

Comments

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

Examples

			Examples for description #1:
For n=2 there are 4 possibilities:
.................
..1..2..12..21...
..2..1...........
.................
For n=3 there are 24 possibilities:
....................
.123..132..231..321.
....................
.1..1..2..3.........
.2..3..3..2.........
.3..2..1..1.........
.........................
.12.21..1.32.3...3.1..23.
..3.3..32..1.21.12.23.1..
.........................
.31.13..1.23.2...2.1..32.
.2...2.23..1.31.13.32.1..
.........................
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.

References

  • A. Baram and D. Kutasov, "On the dynamics of random sequential absorption," J. Physics A: Math. Gen. 22, L251-L254 (1989).
  • Y. Fan and J. K. Percus, Use of Model Solutions in RSA on a Lattice, Phys. Rev. Lett. 67, 1677 (1991).

Crossrefs

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

Extensions

New description and more terms from R. H. Hardin, Oct 25 2003
Entry revised following comments from Joel Yellin (yellin(AT)soe.ucsc.edu), Mar 04 2004

A087783 Array T(n,k) (n >= 1, k >= 1) read by antidiagonals, giving number of ways of arranging the numbers 1 ... mn into an m X n array so there is exactly one local maximum.

Original entry on oeis.org

1, 2, 2, 4, 16, 4, 8, 208, 208, 8, 16, 3584, 29568, 3584, 16, 32, 76544, 7452704, 7452704, 76544, 32, 64, 1947648, 2941306368, 35704394880, 2941306368, 1947648, 64, 128, 57477120, 1683453629440, 331333877743200, 331333877743200, 1683453629440, 57477120, 128, 256, 1929117696, 1323082429842432, 5338455334819710720, 88366736882654697600, 5338455334819710720, 1323082429842432, 1929117696, 256, 512, 72545402880, 1370418864769445888, 137813651152462288749440
Offset: 1

Views

Author

R. H. Hardin, Oct 25 2003

Keywords

Examples

			Array begins:
1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,...
2,16,208,3584,76544,1947648,57477120,1929117696...
4,208,29568,7452704,2941306368,1683453629440...
8,3584,7452704,35704394880,331333877743200,...
16,76544,2941306368,331333877743200,...

Crossrefs

Cf. A000079 (k = 1 or n = 1), A087518 (n=k). A087923-A087932 give rows 2 through 11. Cf. A007846.

Formula

T(n, k) = T(k, n). T(n, 1) = 2^(n-1) (see A000079).

A087519 Number of arrangements of 1..n^2 in n X n array with one local minimum and one local maximum.

Original entry on oeis.org

1, 16, 3584, 58566144
Offset: 1

Views

Author

R. H. Hardin, Oct 23 2003

Keywords

Comments

Neighbors of a cell are only considered in the X and Y directions.

Examples

			For n=2, 12 has one local max. whereas 13 has two local maxima
........ 34 and one local min. ....... 42 and two local minima

Crossrefs

Cf. A087518.
Showing 1-3 of 3 results.

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