For n≥ 0, let Sigma_k,2n denote the component of weight k+1-2n in the Kerov character polynomial Sigma_k, and write [R_2^i]Sigma_k,2n for the coefficient of R_2^i. Stanley's coefficient conjecture. For every i≥ 1, [R_2^i]Sigma_2i+3,4=frac1540i(i+1)^3(i+2)^3(i+3)(2i+3). This conjecture was communicated by Biane as a private conjecture of Stanley concerning higher-order terms. It is proved later in the paper from the stated corollary, so the conjecture is solved.

Stanley's coefficient conjecture for Kerov character polynomials

About 21 years old · traced to 2005

For $n\geq 0$ , let $\Sigma_{k,2n}$ denote the component of weight $k+1-2n$ in the Kerov character polynomial $\Sigma_k$ , and write $[R_2^i]\Sigma_{k,2n}$ for the coefficient of $R_2^i$ . Stanley's coefficient conjecture. For every $i\geq 1$ ,

[R_2^i]\Sigma_{2i+3,4}=\frac{1}{540}i(i+1)^3(i+2)^3(i+3)(2i+3).

This conjecture was communicated by Biane as a private conjecture of Stanley concerning higher-order terms. It is proved later in the paper from the stated corollary, so the conjecture is solved.

References

Primary source

I. P. Goulden and A. Rattan, “An explicit form for Kerov's character polynomials”, arXiv:math/0505317 (2005).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2005 paper gives a proof of the conjecture, and a later paper supplies an independent combinatorial solution.

The conjecture was communicated by Biane as a private conjecture of Stanley concerning higher-order terms in Kerov character polynomials. It asserts an explicit formula for the relevant coefficient.

May 15, 2005 proof

Goulden and Rattan state the conjecture as Conjecture 1.5 and derive it from their Corollary 3.4 by specializing the parameter to $m=2$ , explicitly concluding “Proof of Conjecture 1.5.” Féray and Śniady later gave a combinatorial interpretation that provides an explicit combinatorial solution to the same conjecture.

Current status (as of September 2026): The conjecture has proofs in the Goulden–Rattan paper and in later work by Féray and Śniady; this report classifies the resolution as unverified.

Sources

No solutions have been posted yet.