A247139 The number of tilings of a triangular shape T_n with n rectangles identifying all tilings which use the same rectangular shapes.

1, 1, 2, 3, 6, 11, 23, 45, 95, 195, 417, 875, 1887, 4021, 8727, 18755, 40850, 88366, 192991, 418994, 916490, 1995080, 4367522, 9521434, 20849739, 45493669

Offset: 1

Author

Kival Ngaokrajang, Dec 02 2014

Keywords

• nonn
• more

Comments

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.

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]

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.

From Ludovic Schwob, Mar 25 2025: (Start)

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:

o o

/ \ / \

/ \ / \

/ \ / \

/ \ / \

/ \ / \

/ \ / \

/ \ / \

o \ o \

/ \ \ / \ \

o \ o o \ o

/ \ \ / \ / \ \ / \

o \ \ / \ / \ \ o \

/ \ \ \ / \ / \ \ / \ \

o \ \ \ o o o o \ o \ \

/ \ \ \ \ / \ / \ / \ / \ \ / \ \ \

o o o o o o o o o o o o o o o o o o

These two binary trees are not isomorphic but give tilings which use the same rectangles. (End)

Examples

			For n = 3 (see the Catalan link):
  tiling T1: (3,1), (2,1), (1,1)
  tiling T2: (2,2), (1,1)^2      (self-transposed)
  tiling T3: (1,3), (1,2), (1,1) (transposed of T1)
  tiling T4: (1,3), (2,1), (1,1)
  tiling T5: (3,1), (1,2), (1,1) (transposed of T4)
The total number is 5 = A000108(5) (Catalan).
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.
		

Links

• Kival Ngaokrajang, Illustration for n = 1..6, Illustration for n = 10, Catalan numbers (conjecture)

Crossrefs

Cf. A000108, A002620, A001190.

Extensions

Edited (rewritten name, comments, example) by Wolfdieter Lang, Dec 10 2014

a(12)-a(26) from Ludovic Schwob, Mar 23 2025