[seqfan] Splitting the square (followings - restricted case - link with existing sequences)

Jean-Luc Manguin jean-luc.manguin at unicaen.fr
Thu Nov 19 10:25:49 CET 2020

Hello everyone, 

The sequence [ https://oeis.org/A108066/a108066 | https://oeis.org/A108066/a108066 ] gives the Number of distinct ways to dissect a square into n rectangles of equal area, considering them as "free" items as in the problem of polyominoes. In a preceding post, I proposed a restriction for this sequence, and gave the differences with my results. 
Neil Fernandez proposed a more restricted dissection, as this : you can only cut a rectangle of size 1/n at every step. This leads to much lower numbers in the sequence ; what it interesting, is that this sequence is linked to others in OEIS. 

At first sight, if we consider these dissections as "fixed" items (as for polyominoes) it is easy to come to the formula a(n)= 4*a(n-1) - 2*a(n-2), which is A006012 . 
The main problem in this enumeration is the symmetry ; the results for "free" items are : 

2 1 
3 2 
4 5 
5 13 
6 38 
7 116 
8 372 
9 1220 
10 4072 
11 13712 
12 46448 
13 157840 
14 537440 
15 1832000 
16 6249024 
17 21323840 
18 72780928 

The number of 2-axis symmetrical items s2(n) is A016116 : 1,1,2,2,4,4 etc. with a shift of 2 : s2(n)=A016116(n-2) 

The number of one-axis symmetrical items s1(n) is the sequence A032085, also with a shift s1(n)=A032085(n-1) 

We can notice that s1(n)+s2(n) = 2^(n-2). 

I think the list given above could be a new sequence. Thank you for advices and comments. 

Best regards, 


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