OFFSET
1,2
COMMENTS
Given an m X n chocolate bar, let A(m,n) be the number of ways to break it into m*n unit pieces where each break occurs along a gridline. Order matters, and the pieces are distinguishable. Then this sequence lists the values A(m,n) in increasing order as m and n range over the positive integers.
The sequence of factorials, A000142, is a subsequence as A(1,n) = A(n,1) = (n-1)!.
For m,n>1, A(m,n) is divisible by 2^(m+n-2).
LINKS
Caleb Ji, Tanya Khovanova, Robin Park, Angela Song, Chocolate Numbers, arXiv:1509.06093 [math.CO], 2015.
Caleb Ji, Tanya Khovanova, Robin Park, Angela Song, Chocolate Numbers, Journal of Integer Sequences, Vol. 19 (2016), #16.1.7.
FORMULA
A(m,n)=1 for max(m,n)<2 and A(m,n) = Sum_{i=1..m-1} C(m*n-2,i*n-1) *A(i,n) *A(m-i,n) + Sum_{i=1..n-1} C(m*n-2,i*m-1) *A(m,i) *A(m,n-i) else.
EXAMPLE
For n = m = 2, there are two ways for the first break: breaking it horizontally or vertically. After that we need two more breaks that can be done in any order. Thus A(2,2) = 4, and 4 belongs to the sequence.
MATHEMATICA
terms = 29;
A[m_, n_] := A[m, n] = If[Max[m, n] < 2, 1, Sum[A[i, n] Binomial[m n - 2, i n - 1] A[m - i, n], {i, 1, m - 1}]] + Sum[A[m, i] Binomial[m n - 2, i m - 1] A[m, n - i], {i, 1, n - 1}];
Table[A[m, n], {m, 1, terms}, {n, 1, terms}] // Flatten // Union //
Take[#, terms]&
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved