let-n-be-a-fixed-positive-integer-How-many-ways-are-there-to-write-n-as-a-sum-of-positive-integers-n-a-1-a-2-a-k-with-k-arbitary-positive-integer-and-a-1-a-2-a-k-a-1-1-for-example-wi

Question Number 26513 by gunawan last updated on 26/Dec/17

$$\mathrm{let}\:{n}\:\mathrm{be}\:\mathrm{a}\:\mathrm{fixed}\:\mathrm{positive}\:\mathrm{integer}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{are}\:\mathrm{there}\:\mathrm{to}\:\mathrm{write}\:{n}\:\mathrm{as}\:\mathrm{a}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{positive}\:\mathrm{integers},\: \\ $$$${n}={a}_{\mathrm{1}} +{a}_{\mathrm{2}} +…+{a}_{{k}} \\ $$$$\mathrm{with}\:{k}\:\mathrm{arbitary}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{and}\:{a}_{\mathrm{1}} \leqslant{a}_{\mathrm{2}} …\leqslant{a}_{{k}} \leqslant{a}_{\mathrm{1}} +\mathrm{1}.\:\mathrm{for}\:\mathrm{example} \\ $$$$\mathrm{with}\:{n}=\mathrm{4},\:\mathrm{there}\:\mathrm{are}\:\mathrm{four}\:\mathrm{ways}\::\:\mathrm{4},\:\mathrm{2}+\mathrm{2},\:\mathrm{1}+\mathrm{1}+\mathrm{2},\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1} \\ $$

Commented by mrW1 last updated on 26/Dec/17

Partitions of interger n see: https://www.whitman.edu/mathematics/cgt_online/book/section03.03.html

Commented by gunawan last updated on 26/Dec/17

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much} \\ $$

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