Let f(p;n,k) = C(n,k) p^(k) (1-p)^(n-k)

Definitions and mathematical statement

For $n\ge1$, $0<p<1$, and $0\le k\le n$, let \[ b_{n,p}(k)=\binom nk p^k(1-p)^{n-k} \] be the probability mass function of a binomial random variable. Exclude the symmetric parameter $p=1/2$. Can there exist a single choice of $n,p$ and four pairwise distinct integers $k,\ell,k',\ell'\in\{0,\ldots,n\}$ such that \[ b_{n,p}(k)=b_{n,p}(\ell), \qquad b_{n,p}(k')=b_{n,p}(\ell')\,? \] The two common values need not be equal to one another. The question asks whether two disjoint equality pairs can occur within the same nonsymmetric binomial distribution.

Writing $r=p/(1-p)>0$, an equality for $k<\ell$ is equivalent to \[ r^{\ell-k}=\frac{\binom nk}{\binom n\ell}. \] Thus both pairs must be compatible with the same positive $r\ne1$; satisfying separate equalities with different parameters does not answer the question. The excluded endpoints $p=0,1$ would create many trivial zero masses, and $p=1/2$ produces the usual reflection equalities. Indices at $0$ or $n$ themselves are permitted when $0<p<1$. Resolving the problem means either producing one exact example or proving that all such simultaneous equalities are impossible.

Short English statement

Can one nonsymmetric binomial probability mass function have two equality pairs involving four distinct indices?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3100085 (AMR-030-0085), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Cooper - Combinatorial Problems I Like (2020 snapshot), Probability, source-order bullet 85. Original source indexed by AMR-030-0085. https://people.math.sc.edu/cooper/combprob.html.