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COMMENTS
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Let k = number of splits -- Think of the first split as a bisection of our space, say the unit interval, the second split can be either bisecting the left half or the right half from the first split, etc. Alternatively, think of a "spatial binary branching process" or plane trees.
Let C_k = number of distinct space-partitioning binary trees made from k splits (the Catalan number).
k! = number of ways in which we can reach any of the distinct space-partitioning binary trees attained by k splits, i.e., the total number of distinct paths in the Hasse Diagram representing the split-based transition diagram that is representing the growth of such space-partitioning binary trees.
"Catalan Coefficients" is the number of distinct paths from the unique root tree with 0 splits to each one of the C_k many distinct trees with k splits that are structurally labeled with elements from [C_k] := {0,1,2,...,C_k-1}. Let these distinct path counts from the root tree be our coefficients B_k := (B_{k,0},B_{k,1},...,B_{k,C_k-1}). Note that this recursion for B_k depends on B_{k-1}, i.e., the number of distinct paths from the root tree with 0 splits to each distinct tree j in [C_{k-1}] with k-1 splits and the subset A_i of [C_{k-1}] of trees with k-1 splits that can produce the tree i in [C_k] with k splits. Thus B_k = ( B_{k,0},B_{k,1},...,B_{k,C_k-1} ), B_{k,i} = sum_{j in A_i} B_{k-1,j}, sum_{i=0..C_k-1}B_{k,i}=k!
The following B_k for k in {0,1,2,3,4,5} were hand-computed by Gloria Teng on Aug 21 2009 and verified by Raazesh Sainudiin.
k C_k k! (B_{k,0},B_{k,1},...,B_{k,C_k-1})
0 1 1 1
1 1 1 1
2 2 2 1, 1
3 5 6 1, 1, 2, 1, 1
4 14 24 1, 1, 3, 2, 3, 1, 1, 1, 1, 3, 2, 3, 1, 1
5 42 120 1, 1, 4, 3, 2, 3, 4, 1, 1, 6, 6, 1, 1, 4, 3, 2, 3, 4, 1, 1, 8, 8, 1, 1, 4, 3, 2, 3, 4, 1, 1, 6, 6, 1, 1, 4, 3, 2, 3, 4, 1, 1
6 132 720 ...
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