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.

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

Author

John Tyler Rascoe, Dec 09 2023

Keywords

• nonn
• tabf

Comments

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.

Examples

			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)
		

Links

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

Crossrefs

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

Cf. A005316, A008828, A287220.

Programs

Python

# see linked program