cp's OEIS Frontend

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.

%I A368054 #16 Feb 04 2024 21:23:02
%S A368054 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,
%T A368054 1952,1229,1208,2516,1803,1181,1148,998,478,370,279,125,76,26,13,3,3,
%U A368054 3926,0,3658,21608,17083,17811,48542,51306,40081,51660,59023,42327
%N 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.
%C A368054 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.
%H A368054 Benedict Irwin, <a href="https://doi.org/10.22541/au.162022396.68662845/v1">On the Number of k-Crossing Partitions</a>, Univ. of Cambridge (2021).
%H A368054 John Tyler Rascoe, <a href="/A368054/a368054_1.py.txt">Python program</a>.
%e A368054 Triangle begins:
%e A368054        k=0  1   2    3   4   5   6   7   8   9  10  11  12
%e A368054   n=0:   1;
%e A368054   n=1:   1;
%e A368054   n=2:   3, 0,  1;
%e A368054   n=3:  14, 0,  8,  10,  2,  2;
%e A368054   n=4:  81, 0, 59, 162, 70, 66, 82, 22, 19,  6,  7,  0,  2;
%e A368054   ...
%e A368054 Row n = 3 counts the following k-crossing partitions.
%e A368054 T(3,0) = 14:   T(3,2) = 8:    T(3,3) = 10:   T(3,4) = 2:    T(3,5) = 2:
%e A368054 (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)
%e A368054 (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)
%e A368054 (1,2,5,4,3,6)  (3,6,5,4,1,2)  (1,4,5,2,3,6)
%e A368054 (1,4,3,2,5,6)  (5,2,1,6,3,4)  (1,6,3,2,5,4)
%e A368054 (1,4,5,6,3,2)  (5,4,3,6,1,2)  (3,2,5,4,1,6)
%e A368054 (1,6,3,4,5,2)  (5,6,1,2,3,4)  (3,4,1,2,5,6)
%e A368054 (1,6,5,2,3,4)  (5,6,1,4,3,2)  (3,6,5,2,1,4)
%e A368054 (1,6,5,4,3,2)  (5,6,3,2,1,4)  (5,2,1,4,3,6)
%e A368054 (3,2,1,4,5,6)                 (5,4,1,6,3,2)
%e A368054 (3,2,1,6,5,4)                 (5,6,3,4,1,2)
%e A368054 (3,4,5,2,1,6)
%e A368054 (5,2,3,4,1,6)
%e A368054 (5,4,1,2,3,6)
%e A368054 (5,4,3,2,1,6)
%o A368054 (Python) # see linked program
%Y A368054 Cf. A077054 (column k=0), A001044 (row sums).
%Y A368054 Cf. A005316, A008828, A287220.
%K A368054 nonn,tabf
%O A368054 0,3
%A A368054 _John Tyler Rascoe_, Dec 09 2023