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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.
1
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
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).
Sequence in context: A135313 A322670 A277410 * A289546 A334823 A279031
KEYWORD
nonn,tabf
AUTHOR
John Tyler Rascoe, Dec 09 2023
STATUS
approved