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%I #36 Apr 06 2020 22:23:02
%S 1,1,2,1,4,1,4,12,18,12,8,6,2,44,56,120,28,88,4,36,0,8
%N 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.
%C 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.
%C This entry is motivated by A258643.
%C For bisymmetric matrices see the Wikipedia link.
%C 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.
%C 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.
%C 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.
%C 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.
%C For the row sums see A261244.
%H Kival Ngaokrajang, <a href="/A261242/a261242.pdf">Illustration of T(n,k) for n = 1..5, k >= 0</a>, <a href="/A261242/a261242_1.pdf">T(6,0)</a>, <a href="/A261242/a261242_2.pdf">T(6,1)</a>, <a href="/A261242/a261242_3.pdf">T(6,2)</a>, <a href="/A261242/a261242_4.pdf">T(6,4)</a>, <a href="/A261242/a261242_5.pdf">T(6,k) for k = 3, 5, 6, 8</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bisymmetric_matrix">Bisymmetric Matrix</a>.
%e The irregular triangle T(n, k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 ...
%e 1: 1
%e 2: 1
%e 3: 2 1
%e 4: 4 1 4
%e 5: 12 18 12 8 6 2
%e 6: 44 56 120 28 88 4 36 0 8
%e ...
%e n=4: k=0:
%e [[1,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]],
%e [[1,0,0,1], [0,1,1,0], [0,1,1,0], [1,0,0,1]],
%e [[1,1,0,1], [1,1,1,0], [0,1,1,1], [1,0,1,1]],
%e [[1,0,1,1], [0,1,1,1], [1,1,1,0], [1,1,0,1]];
%e k=1:
%e [[1,1,1,1], [1,0,0,1], [1,0,0,1], [1,1,1,1]];
%e k=2:
%e [[1,1,1,1], [1,0,1,1], [1,1,0,1], [1,1,1,1]],
%e [[1,1,1,1], [1,1,0,1], [1,0,1,1], [1,1,1,1]],
%e [[1,1,0,1], [1,0,1,0], [0,1,0,1], [1,0,1,1]],
%e [[1,0,1,1], [0,1,0,1], [1,0,1,0], [1,1,0,1]].
%Y Cf. A000982, A002620, A014616, A258643, A261243, A261244.
%K nonn,tabf,more
%O 1,3
%A _Wolfdieter Lang_ and _Kival Ngaokrajang_, Aug 18 2015
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