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

0,3

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.

Links

Table of n, a(n) for n=0..55.

Benedict Irwin, On the Number of k-Crossing Partitions, Univ. of Cambridge (2021).

John Tyler Rascoe, Python program.

Example

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)

Prog

(Python) # see linked program

Crossrefs

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

Cf. A005316, A008828, A287220.

Sequence in context: A135313 A322670 A277410 * A289546 A334823 A279031

Adjacent sequences: A368051 A368052 A368053 * A368055 A368056 A368057

Keyword

nonn,tabf

Author

John Tyler Rascoe, Dec 09 2023

Status

approved