A261242 Irregular triangle T(n, k) of number of connected bisymmetric n X n matrices B_n with 0 or 1 entries, B_n[1,1] = 1 = B_n[1,n], and k islands of 0's.

1, 1, 2, 1, 4, 1, 4, 12, 18, 12, 8, 6, 2, 44, 56, 120, 28, 88, 4, 36, 0, 8

Offset: 1

Author

Wolfdieter Lang and Kival Ngaokrajang, Aug 18 2015

Keywords

• nonn
• tabf
• more

Comments

The row length sequence is 1 for n = 1 and A000982(n-2) + 1 for n >= 2, that is: 1, 1, 2, 3, 6, 9, 14, 19, 26, 33, 42, ... = A261243.

This entry is motivated by A258643.

For bisymmetric matrices see the Wikipedia link.

For the number of independent entries of an n X n bisymmetric matrix B_n see a Jul 07 2015 comment on A002620(n+1), n >= 1. For the binary case (only 0 and 1 entries) see A060656(n+1), and the Dennis P. Walsh comment and link. If B_n[1,1] and B_n[1,n] is given then the four corners are fixed, and, for n >= 3, there are A002620(n+1) - 2 = A014616(n-2) entries free.

If the n X n bisymmetric matrix B_n of 0's and 1's with B_n[1, 1] = 1 = B_n[1, n] is considered as a grid of n^2 squares of length 1 (in some length unit) with the four corners filled with 1's and the other squares with 0 or 1 then a path between the centers of squares with step length 1 can be defined. No diagonal steps (length sqrt(2)) are allowed. B_n is called connected if there exists no path of 0's which dissects the grid into two parts.

An island of 0's (a 0-island) in B_n is defined as a set of 0's for which each pair is connected by a path of 0's, and a 0 entry at the coast of a 0-island has at least one entry 1 one step away. A single square filled with a 0 is a 0-island if all four neighbors 1 step (of length 1) apart are filled with 1's. If k=0 there exists no such 0-island. See the n=4 examples with k >=1 below. The k = 1 matrix has one simply connected 0-island of four squares. The four k = 2 matrices have two 0-islands consisting of one square each.

See the link with the figures by K. N. where red squares stand for 1 and empty squares for 0. Each matrix appears there rotated by 45 degrees in the counterclockwise direction. The mirror operation means row reversion in the matrix B_n. In the figures this is a mirror operation w.r.t. the middle NW-SE diagonal. 0-islands appear in the figures as holes.

For the row sums see A261244.

Examples

			The irregular triangle T(n, k) begins:
n\k   0   1    2   3   4  5   6   7   8  ...
1:    1
2:    1
3:    2   1
4:    4   1    4
5:   12  18   12   8   6  2
6:   44  56  120  28  88  4  36   0   8
...
n=4: k=0:
[[1,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]],
[[1,0,0,1], [0,1,1,0], [0,1,1,0], [1,0,0,1]],
[[1,1,0,1], [1,1,1,0], [0,1,1,1], [1,0,1,1]],
[[1,0,1,1], [0,1,1,1], [1,1,1,0], [1,1,0,1]];
     k=1:
[[1,1,1,1], [1,0,0,1], [1,0,0,1], [1,1,1,1]];
     k=2:
[[1,1,1,1], [1,0,1,1], [1,1,0,1], [1,1,1,1]],
[[1,1,1,1], [1,1,0,1], [1,0,1,1], [1,1,1,1]],
[[1,1,0,1], [1,0,1,0], [0,1,0,1], [1,0,1,1]],
[[1,0,1,1], [0,1,0,1], [1,0,1,0], [1,1,0,1]].
		

Links

• Kival Ngaokrajang, Illustration of T(n,k) for n = 1..5, k >= 0, T(6,0), T(6,1), T(6,2), T(6,4), T(6,k) for k = 3, 5, 6, 8
• Wikipedia, Bisymmetric Matrix.

Crossrefs

Cf. A000982, A002620, A014616, A258643, A261243, A261244.