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p(10) = 42
42
The partition function at n = 10. Forty-two unordered sums of positive integers.
Catalan 42 asks how relations can be legally arranged: parentheses that close, paths that stay above the ground. Partition 42 asks how a whole can be split when order is forgotten. Ten written as a sum of positive integers, disregarding order, in forty-two ways. 6+4 is the same pile as 4+6.
What it supports
An exact value of p(n). AIM prints p(10)=42 as a fact one could check by hand. OEIS A000041 is 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42. NIST DLMF §27.14 names the same integer. Euler’s generating function ∏ 1/(1−xᵐ) has that coefficient. Two combinatorial 42s. Two questions.
What it does not
Not Catalan C₅. Not the cake number. Not a measured 42. Order does not count: 3+7 and 7+3 are one partition.
Calculation
p(10)=42. Sequence: 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42. Examples: 10; 9+1; 8+2; 7+3; 7+2+1; 5+5; 4+3+2+1; 1+1+1+1+1+1+1+1+1+1. Order forgotten.
Unrestricted integer partitions. Order disregarded.
ExactnessExact
Form42
Unitpartitions
Strengthhard
Hung9 September 2026