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%I #52 Sep 01 2018 17:41:48
%S 1,1,2,3,6,11,23,45,95,195,417
%N The number of tiling of a triangular shape T_n with n rectangles identifying all tilings which use the same rectangular shapes.
%C The triangular shape T_n has here row length k for row k, for k = 1, 2, ..., n. Consider tilings of T_n with n rectangular shapes (i,j),(i < j, i > j, i=j (square)). The number of possible rectangular shapes (i,j) with n >= i >= j >= 1, which can show up at all in the tiling of T_n (together with their transposed shapes with i interchanged with j) is given as A002620(n+1). See the Dec 09 2014 comment and example by _Wolfdieter Lang_ there.
%C The number of such tilings is conjectures to be A000108(n) (Catalan numbers). See the examples in the Catalan number link. For each tiling there is a transposed tiling obtained by mirroring w.r.t. the antidiagonal (SW-NE). A tiling may be self-transposed. E.g., n = 4 has transposed pairs a, m; b, h; x, i; c, g; d, j; e, k; and f, e. Tiling a is (4,1), (3,1), (2,1), (1,1), tiling m is then (1,4), (1,3), (1,2), (1,1).
%C For the present sequence not only the transposed tilings are identified but all other tilings with the same rectangular shapes irrespective of the order in (i,j). Thus, in the Catalan link a and m and d and j and e and k and f and e are all identified. E.g., tiling j has tiles (4,1), (1,3), (1.2), (1,1) which is, by taking (1,3) as (3,1), and (1,2) as (2,1) the same as tiling a. Tilings c and g are identified with tiling b. Therefore, for n=4 there remain only 3 tilings with representatives a, b and x given in the link. They represent 8, 4 and 2 tilings, respectively.
%H Kival Ngaokrajang, <a href="/A247139/a247139_1.pdf">Illustration for n = 1..6</a>, <a href="/A247139/a247139_2.pdf">Illustration for n = 10</a>, <a href="/A247139/a247139_3.pdf">Catalan numbers (conjecture)</a>
%e For n = 3 (see the Catalan link):
%e tiling T1: (3,1), (2,1), (1,1)
%e tiling T2: (2,2), (1,1)^2 (self-transposed)
%e tiling T3: (1,3), (1,2), (1,1) (transposed of T1)
%e tiling T4: (1,3), (2,1), (1,1)
%e tiling T5: (3,1), (1,2), (1,1) (transposed of T4)
%e The total number is 5 = A000108(5) (Catalan).
%e T3 is identified with T1 by taking (1,3) as (3,1) and (1,2) as (2,1). Similarly, T4 and T5 are also identified with T1. Representatives are T1 and T2, representing 4 and 1 tilings,respectively. Therefore a(3) = 2.
%Y Cf. A000108, A002620.
%K nonn,more
%O 1,3
%A _Kival Ngaokrajang_, Dec 02 2014
%E Edited (rewritten name, comments, example) by _Wolfdieter Lang_, Dec 10 2014
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