A287220 -id:A287220 - cp's OEIS frontend

A287221 Number of twice-crossing partitions on n nodes.

0, 0, 0, 0, 8, 42, 168, 760, 2418, 10490, 30842, 131676

Offset: 0

Benedict W. J. Irwin, May 22 2017

The meandric numbers A005316 are the numbers of paths which cross themselves 0 times.

This sequence is the number of paths that must cross themselves exactly twice.

			a(4) = 8; this is from the partitions
(2,1,4,3), (2,4,3,1),
(3,2,4,1), (3,4,2,1),
(4,1,3,2), (4,2,1,3),
(4,2,3,1), (4,3,1,2).

B. W. J. Irwin, On the number of k-crossing partitions

Cf. A005316, A287220.

A368054 Irregular triangle read by rows: T(n,k) is the number of k-crossing partitions on 2n nodes, where all partition terms alternate in parity, counted up to reflection.

Original entry on oeis.org

1, 1, 3, 0, 1, 14, 0, 8, 10, 2, 2, 81, 0, 59, 162, 70, 66, 82, 22, 19, 6, 7, 0, 2, 538, 0, 454, 1952, 1229, 1208, 2516, 1803, 1181, 1148, 998, 478, 370, 279, 125, 76, 26, 13, 3, 3, 3926, 0, 3658, 21608, 17083, 17811, 48542, 51306, 40081, 51660, 59023, 42327

Offset: 0

John Tyler Rascoe, Dec 09 2023

The 0-crossing partitions counted in A005316 all have terms that alternate in parity. Also, for an even number of nodes the partitions 1432 and 2341 count the same meandric path. This triangle aims to reduce the total number of k-crossing partitions considered from (2*n)! to (n!)^2, see Irwin link.

			Triangle begins:
       k=0  1   2    3   4   5   6   7   8   9  10  11  12
  n=0:   1;
  n=1:   1;
  n=2:   3, 0,  1;
  n=3:  14, 0,  8,  10,  2,  2;
  n=4:  81, 0, 59, 162, 70, 66, 82, 22, 19,  6,  7,  0,  2;
  ...
Row n = 3 counts the following k-crossing partitions.
T(3,0) = 14:   T(3,2) = 8:    T(3,3) = 10:   T(3,4) = 2:    T(3,5) = 2:
(1,2,3,4,5,6)  (3,4,1,6,5,2)  (1,2,5,6,3,4)  (3,2,5,6,1,4)  (3,6,1,4,5,2)
(1,2,3,6,5,4)  (3,4,5,6,1,2)  (1,4,3,6,5,2)  (3,6,1,2,5,4)  (5,2,3,6,1,4)
(1,2,5,4,3,6)  (3,6,5,4,1,2)  (1,4,5,2,3,6)
(1,4,3,2,5,6)  (5,2,1,6,3,4)  (1,6,3,2,5,4)
(1,4,5,6,3,2)  (5,4,3,6,1,2)  (3,2,5,4,1,6)
(1,6,3,4,5,2)  (5,6,1,2,3,4)  (3,4,1,2,5,6)
(1,6,5,2,3,4)  (5,6,1,4,3,2)  (3,6,5,2,1,4)
(1,6,5,4,3,2)  (5,6,3,2,1,4)  (5,2,1,4,3,6)
(3,2,1,4,5,6)                 (5,4,1,6,3,2)
(3,2,1,6,5,4)                 (5,6,3,4,1,2)
(3,4,5,2,1,6)
(5,2,3,4,1,6)
(5,4,1,2,3,6)
(5,4,3,2,1,6)

Benedict Irwin, On the Number of k-Crossing Partitions, Univ. of Cambridge (2021).
John Tyler Rascoe, Python program.

Cf. A077054 (column k=0), A001044 (row sums).

Cf. A005316, A008828, A287220.

Python
```
# see linked program
```

A287222 Number of 3-time self-crossing partitions on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 2, 16, 164, 944, 4386, 22240, 83066, 398132

Offset: 0

Benedict W. J. Irwin, May 22 2017

The meandric numbers A005316 are the numbers of paths which cross themselves 0 times.

This sequence is the number of paths that must cross themselves exactly 3 times.

			a(4) = 2, this is from the partitions (2,4,1,3) and (3,4,1,2).

B. W. J. Irwin, On the number of k-crossing partitions

Cf. A005316, A287220, A287221.

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A287221 Number of twice-crossing partitions on n nodes.

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A368054 Irregular triangle read by rows: T(n,k) is the number of k-crossing partitions on 2n nodes, where all partition terms alternate in parity, counted up to reflection.

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Python

A287222 Number of 3-time self-crossing partitions on n nodes.

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