A360916 -id:A360916 - cp's OEIS frontend

A360199 Array read by antidiagonals: T(m,n) is the number of induced paths in the grid graph P_m X P_n.

0, 1, 1, 3, 8, 3, 6, 25, 25, 6, 10, 58, 94, 58, 10, 15, 117, 270, 270, 117, 15, 21, 218, 681, 1004, 681, 218, 21, 28, 387, 1597, 3330, 3330, 1597, 387, 28, 36, 666, 3592, 10224, 14864, 10224, 3592, 666, 36, 45, 1123, 7880, 29924, 61165, 61165, 29924, 7880, 1123, 45

Offset: 1

Andrew Howroyd, Jan 29 2023

Paths of length zero are not counted here.

			Array begins:
============================================================
m\n|  1   2    3     4      5       6        7         8 ...
---+--------------------------------------------------------
1  |  0   1    3     6     10      15       21        28 ...
2  |  1   8   25    58    117     218      387       666 ...
3  |  3  25   94   270    681    1597     3592      7880 ...
4  |  6  58  270  1004   3330   10224    29924     85036 ...
5  | 10 117  681  3330  14864   61165   238897    907148 ...
6  | 15 218 1597 10224  61165  334536  1723535   8647932 ...
7  | 21 387 3592 29924 238897 1723535 11546874  75134416 ...
8  | 28 666 7880 85036 907148 8647932 75134416 629381852 ...
   ...

Andrew Howroyd, Table of n, a(n) for n = 1..435
Eric Weisstein's World of Mathematics, Grid Graph.
Wikipedia, Induced path.

Main diagonal is A360200.
Rows 1..2 are A000217(n-1), A360201.
Cf. A287151 (induced connected subgraphs), A288518 (undirected paths), A360196 (induced cycles), A360202 (induced trees), A360916 (maximum induced paths).

A331986 Number of snake-like polyominoes with the maximum possible number of unit squares in an n X n square.

Original entry on oeis.org

1, 4, 8, 84, 56, 136, 52, 216, 16, 1504, 2352, 1152, 1344, 123216, 82432, 11008, 308992

Offset: 1

Alain Goupil, Feb 03 2020

The maximum possible number of unit squares is given by A331968(n).
Equivalently, a(n) is the number of maximum length paths without chords in the n X n grid graph. A path without chords is an induced subgraph that is a path.
For n > 1, a(n) is a multiple of 4 since a solution can have at most one symmetry considering rotations and reflections. - Andrew Howroyd, Feb 04 2020

			For n = 4 the number of snake-like polyominoes with 11 cells is 84.

Eric Weisstein's World of Mathematics, Grid Graph

Main diagonal of A360916.
Cf. A331968, A059525 (connected induced subgraphs), A099155.
Cf. A332920 (non-isomorphic snakes), A332921 (symmetric snakes).

a(15) from Andrew Howroyd, Feb 04 2020

a(16)-a(17) from Yi Yang, Oct 03 2022

A360918 Array read by antidiagonals: T(m,n) is the number of maximum induced trees in the grid graph P_m X P_n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 10, 10, 10, 1, 1, 4, 26, 26, 4, 1, 1, 24, 2, 32, 2, 24, 1, 1, 8, 10, 50, 50, 10, 8, 1, 1, 56, 50, 56, 22, 56, 50, 56, 1, 1, 16, 194, 64, 608, 608, 64, 194, 16, 1, 1, 128, 4, 72, 182, 4120, 182, 72, 4, 128, 1, 1, 32, 32, 80, 16, 208, 208, 16, 80, 32, 32, 1

Offset: 1

Andrew Howroyd, Feb 26 2023

A maximum induced tree is an induced tree of greatest size.

			Array begins:
===============================================
m\n| 1  2   3  4   5     6    7     8     9 ...
---+-------------------------------------------
1  | 1  1   1  1   1     1    1     1     1 ...
2  | 1  4   2 10   4    24    8    56    16 ...
3  | 1  2  10 26   2    10   50   194     4 ...
4  | 1 10  26 32  50    56   64    72    80 ...
5  | 1  4   2 50  22   608  182    16     2 ...
6  | 1 24  10 56 608  4120  208  1968 22716 ...
7  | 1  8  50 64 182   208  488   560  1050 ...
8  | 1 56 194 72  16  1968  560 65864 14340 ...
9  | 1 16   4 80   2 22716 1050 14340   166 ...
   ...

Andrew Howroyd, Table of n, a(n) for n = 1..435
Eric Weisstein's World of Mathematics, Grid Graph.

Main diagonal is A360919.

Cf. A360202, A360847, A360913, A360916, A360920 (maximum sizes).

T(m,n) = T(n,m).

A360913 Array read by antidiagonals: T(m,n) is the number of maximum induced cycles in the grid graph P_m X P_n.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 1, 1, 4, 5, 1, 7, 1, 5, 6, 1, 17, 17, 1, 6, 7, 1, 31, 90, 31, 1, 7, 8, 1, 8, 284, 284, 8, 1, 8, 9, 1, 40, 9, 1398, 9, 40, 1, 9, 10, 1, 120, 163, 10, 10, 163, 120, 1, 10, 11, 1, 8, 1329, 382, 23, 382, 1329, 8, 1, 11, 12, 1, 64, 128, 6098, 190, 190, 6098, 128, 64, 1, 12

Offset: 2

Andrew Howroyd, Feb 26 2023

A maximum induced cycle is an induced cycle of longest length.

			Array begins:
=========================================
m\n| 2  3   4    5    6   7    8    9 ...
---+-------------------------------------
2  | 1  2   3    4    5   6    7    8 ...
3  | 2  1   1    1    1   1    1    1 ...
4  | 3  1   7   17   31   8   40  120 ...
5  | 4  1  17   90  284   9  163 1329 ...
6  | 5  1  31  284 1398  10  382 6098 ...
7  | 6  1   8    9   10  23  190  388 ...
8  | 7  1  40  163  382 190 2100   80 ...
9  | 8  1 120 1329 6098 388   80 6840 ...
  ...

Eric Weisstein's World of Mathematics, Grid Graph.

Main diagonal is A360914.

Cf. A360196 (induced cycles), A360915 (lengths), A360916, A360918.

T(m,n) = T(n,m).

A360917 Array read by antidiagonals: T(m,n) is the number of vertices in the longest induced path in the grid graph P_m X P_n.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 5, 5, 4, 5, 6, 7, 6, 5, 6, 8, 9, 9, 8, 6, 7, 9, 11, 11, 11, 9, 7, 8, 11, 14, 14, 14, 14, 11, 8, 9, 12, 16, 17, 17, 17, 16, 12, 9, 10, 14, 18, 20, 21, 21, 20, 18, 14, 10, 11, 15, 20, 22, 24, 24, 24, 22, 20, 15, 11, 12, 17, 22, 25, 27, 29, 29, 27, 25, 22, 17, 12

Offset: 1

Andrew Howroyd, Feb 26 2023

Equivalently, T(m,n) is the maximum number of unit squares of a snake-like polyomino in an m X n rectangle.

			Array begins:
==============================================
  m\n|  1  2  3  4  5  6  7  8  9 10 11 12 ...
-----+----------------------------------------
   1 |  1  2  3  4  5  6  7  8  9 10 11 12 ...
   2 |  2  3  5  6  8  9 11 12 14 15 17 18 ...
   3 |  3  5  7  9 11 14 16 18 20 22 24 26 ...
   4 |  4  6  9 11 14 17 20 22 25 28 30 33 ...
   5 |  5  8 11 14 17 21 24 27 30 34 37 40 ...
   6 |  6  9 14 17 21 24 29 32 36 40 44 47 ...
   7 |  7 11 16 20 24 29 33 38 42 46 50 55 ...
   8 |  8 12 18 22 27 32 38 42 48 52 57 62 ...
   9 |  9 14 20 25 30 36 42 48 53 58 64 70 ...
  10 | 10 15 22 28 34 40 46 52 58 64 71 77 ...
  11 | 11 17 24 30 37 44 50 57 64 71 77 86 ...
  12 | 12 18 26 33 40 47 55 62 70 77 86 92 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..435
Nikolai Beluhov, Snake paths in king and knight graphs, arXiv:2301.01152 [math.CO], 2023.
Eric Weisstein's World of Mathematics, Grid Graph.

Main diagonal is A331968.

Cf. A360199, A360915, A360916 (maximum induced paths), A360920.

T(m,n) = T(n,m).

T(m,n) = 2*m*n/3 + O(m+n) (Beluhov 2023, Proposition 3). - Pontus von Brömssen, May 08 2023

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A360199 Array read by antidiagonals: T(m,n) is the number of induced paths in the grid graph P_m X P_n.

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A331986 Number of snake-like polyominoes with the maximum possible number of unit squares in an n X n square.

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A360918 Array read by antidiagonals: T(m,n) is the number of maximum induced trees in the grid graph P_m X P_n.

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A360913 Array read by antidiagonals: T(m,n) is the number of maximum induced cycles in the grid graph P_m X P_n.

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A360917 Array read by antidiagonals: T(m,n) is the number of vertices in the longest induced path in the grid graph P_m X P_n.

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