The truncated pentagonal number theorem

Binary Diagrams for Storing Ascending Compositions

“…Taking into account that p (2) (n) ≤ p(n), by Theorem 4.1 we deduce that Algorithm 2 performs fewer assignment statements than Algorithm 5 in [7,Section 4]. Both algorithms evaluate the same number of logical expressions.…”

“…Algorithm 2 executes 3p(n) + 5p (2) (n) assignment statements and evaluates p(n) + 3p (2) (n) boolean expressions.…”

“…1 we can see that (1, 5)(1, 4)(1, 3) (2, 2)(2, 0) is a path that connects the root node (1,5) to the leaf node (2,0). From this path, node (1, 3) is deleted when listing because it is followed by the node (2, 2), which is its right child.…”

“…(2) To generate all the ascending compositions of n, we can traverse in depth-first order the partition strict binary tree. When we reach a leaf node, we list from the path that connects the root node with the leaf node only the leaf node and the nodes that are followed by the left child.…”

“…From this path, node (1, 3) is deleted when listing because it is followed by the node (2, 2), which is its right child. Keeping from every remained pair only the first value, we get the ascending composition [1,1,2,2].…”