Add "partition number" `p(n)`

[inkydragon]

partition_num(n::Int) == length(integer_partitions(n::Int)) == length(partitions(n::Int))

Clean and export npartitions

Combinatorics.jl/src/partitions.jl

Lines 105 to 124 in edb9c2c

	let _npartitions = Dict{Int,Int}()
	global npartitions
	function npartitions(n::Int)
	if n < 0
	0
	elseif n < 2
	1
	elseif (np = get(_npartitions, n, 0)) > 0
	np
	else
	np = 0
	sgn = 1
	for k = 1:n
	np += sgn * (npartitions(n - (k*(3k-1)) >> 1) + npartitions(n - (k*(3k+1)) >> 1))
	sgn = -sgn
	end
	_npartitions[n] = np
	end
	end
	end

ref:

• number of partitions of n - A000041 - OEIS
• NrPartitions(n) - GAP (ref) - Chapter 16: Combinatorics
• Partition Function P - MathWorld
• PartitionsP[n] — Wolfram