This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A247139 #73 Apr 04 2025 22:45:43 %S A247139 1,1,2,3,6,11,23,45,95,195,417,875,1887,4021,8727,18755,40850,88366, %T A247139 192991,418994,916490,1995080,4367522,9521434,20849739,45493669 %N A247139 The number of tilings of a triangular shape T_n with n rectangles identifying all tilings which use the same rectangular shapes. %C A247139 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 A247139 The number of such tilings is equal to 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). [There is a bijection with binary trees. - _Ludovic Schwob_, Mar 23 2025] %C A247139 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. %C A247139 From _Ludovic Schwob_, Mar 25 2025: (Start) %C A247139 a(n) is the number of binary trees with n internal nodes up to isomorphism (up to exchange of left and right children of each internal node) and up to exchange of subtrees with the same number of nodes. Isomorphic binary trees are equivalent, but the converse is not true: %C A247139 o o %C A247139 / \ / \ %C A247139 / \ / \ %C A247139 / \ / \ %C A247139 / \ / \ %C A247139 / \ / \ %C A247139 / \ / \ %C A247139 / \ / \ %C A247139 o \ o \ %C A247139 / \ \ / \ \ %C A247139 o \ o o \ o %C A247139 / \ \ / \ / \ \ / \ %C A247139 o \ \ / \ / \ \ o \ %C A247139 / \ \ \ / \ / \ \ / \ \ %C A247139 o \ \ \ o o o o \ o \ \ %C A247139 / \ \ \ \ / \ / \ / \ / \ \ / \ \ \ %C A247139 o o o o o o o o o o o o o o o o o o %C A247139 These two binary trees are not isomorphic but give tilings which use the same rectangles. (End) %H A247139 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 A247139 For n = 3 (see the Catalan link): %e A247139 tiling T1: (3,1), (2,1), (1,1) %e A247139 tiling T2: (2,2), (1,1)^2 (self-transposed) %e A247139 tiling T3: (1,3), (1,2), (1,1) (transposed of T1) %e A247139 tiling T4: (1,3), (2,1), (1,1) %e A247139 tiling T5: (3,1), (1,2), (1,1) (transposed of T4) %e A247139 The total number is 5 = A000108(5) (Catalan). %e A247139 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 A247139 Cf. A000108, A002620, A001190. %K A247139 nonn,more %O A247139 1,3 %A A247139 _Kival Ngaokrajang_, Dec 02 2014 %E A247139 Edited (rewritten name, comments, example) by _Wolfdieter Lang_, Dec 10 2014 %E A247139 a(12)-a(26) from _Ludovic Schwob_, Mar 23 2025