A partition lambda=(lambda_1,lambda_2,…,lambda_ell) is gap-free, or compact, when 0≤lambda_i-lambda_i+1≤ 1 for all 1≤ i≤ell-1. Let a_2(n) denote the number of gap-free partitions of n. Beck's conjecture. a_2(n) is also the sum of the smallest parts in the distinct partitions of n with an odd number of parts. The paper attributes the generating function for a_2(n) to Andrews and develops combinatorial results concerning this conjecture, but the supplied material does not state that the conjecture has been resolved.

Beck's gap-free partition sum conjecture

George Beck conjectured that, for every positive integer $n$ , the number of compact partitions of $n$ equals the sum of the smallest parts in distinct partitions of $n$ having an odd number of parts.

Known results

Shishuo Fu and Dazhao Tang proved the identity for all $n\geq 1$ by equating the two generating functions; the work appeared in 2018 after its 2017 preprint.
George E. Andrews independently stated and proved the stronger identity $a(n)=b(n)=c(n)$ using generating functions and differentiation.

2017–2018 proof

Fu and Tang’s theorem is presented as a proof of Beck’s conjecture, and Andrews’s paper gives an independent proof. Both sources note that finding bijective or combinatorial proofs would remain interesting; this concerns the proof method, not the truth of the identity.

Current status (as of August 2026): Beck’s identity is resolved by published generating-function proofs, while a bijective or purely combinatorial proof remains open.

Beck's gap-free partition sum conjecture

Primary source

Progress summary

Solutions 0