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Staircase conjecture for first-occurrence layers in the partition graph

Let Lr(n)L_r(n) be the vertices of local simplex dimension rr in the partition graph, let τ(r)\tau(r) be the least nn for which Lr(n)L_r(n) is nonempty, and let δr=(r,r−1,…,2,1)\delta_r=(r,r-1,\dots,2,1) be the staircase partition. An addable corner is a cell that can be added to a Young diagram while preserving its partition shape. Staircase first-occurrence conjecture. For every r≥2r\ge 2, the first-occurrence layer

Lr(τ(r))L_r(\tau(r))

consists exactly of the partitions obtained from δr\delta_r by adding one cell at an addable corner. Computations verify this description for 2≤r≤72\le r\le 7; the assertion for all rr remains open.

Progress summary

Solved

A reader-written complete-proof claim has appeared, but it has not been independently checked, so the conjecture is not resolved.

Fedor B. Lyudogovskiy formulated the conjecture in 2026: the first layer of local dimension rr should occur at the staircase size and consist exactly of its one-cell extensions, for every r≥2r\ge 2.

Known results

  • Exhaustive computation through n≤30n\le 30 verifies the predicted layers for 2≤r≤72\le r\le 7.
  • The computed thresholds are τ(r)=1+r(r+1)/2\tau(r)=1+r(r+1)/2 for 2≤r≤72\le r\le 7.
  • The verified layers have cardinality r+1r+1; for example, L2(4)={(3,1),(2,2),(2,1,1)}L_2(4)=\{(3,1),(2,2),(2,1,1)\}.
  • The paper proves local star/top capacity criteria, but explicitly does not derive the global first-occurrence-family statement.

Posted attempt

A reader-written argument claims a complete proof for every r≥2r\ge 2, deriving τ(r)=r(r+1)/2+1\tau(r)=r(r+1)/2+1 and exactly the r+1r+1 one-cell staircase extensions. This attempt has not been independently verified.

Current status (as of August 2026): The cases 2≤r≤72\le r\le 7 are computationally settled, while a complete proof has been posted but remains unverified and the all-rr conjecture is therefore not established.

Sources
Sources & referencesView supporting material

Primary source

Fedor B. Lyudogovskiy, “Simplex Stratification and Phase Boundaries in the Partition Graph”, arXiv:2603.23228 (2026).

integer-partitionspartition-graphssimplicial-complexes

Solutions 1

Proof

Proof of the complete staircase first-occurrence family

The previously established local corner formulas in arXiv:2603.18696, Corollary 3.4, Proposition 3.7, and Theorem 7.1, are used as known input.

Write

λ=(d1m1,…,dkmk),d1>⋯>dk>0,gi=di−di+1,dk+1=0.\lambda=(d_1^{m_1},\ldots,d_k^{m_k}), \qquad d_1>\cdots>d_k>0, \qquad g_i=d_i-d_{i+1}, \qquad d_{k+1}=0.

For removable corner ii and addable corner jj, the known corner-incidence matrix is

Mij=1−1{j=i, mi=1}−1{j=i+1, gi=1}.M_{ij} = 1-\mathbf1_{\{j=i,\ m_i=1\}} -\mathbf1_{\{j=i+1,\ g_i=1\}}.

Hence the star capacities are

si=k+1−1{mi=1}−1{gi=1},s_i=k+1-\mathbf1_{\{m_i=1\}}-\mathbf1_{\{g_i=1\}},

and every top capacity is at most kk. Consequently

dim⁡loc(λ)={0,λ=(1),k−1,λ=(k,k−1,…,1), k≥2,k+1,mi≥2 and gi≥2 for some i,k,otherwise.(1)\dim_{\mathrm{loc}}(\lambda)= \begin{cases} 0,&\lambda=(1),\\ k-1,&\lambda=(k,k-1,\ldots,1),\ k\ge2,\\ k+1,&m_i\ge2\text{ and }g_i\ge2\text{ for some }i,\\ k,&\text{otherwise}. \end{cases} \tag{1}

Put ai=mi−1a_i=m_i-1, bi=gi−1b_i=g_i-1, and Tk=k(k+1)/2T_k=k(k+1)/2. Direct summation gives the exact weight identity

∣λ∣=Tk+∑i=1k(k−i+1)ai+∑i=1kibi+∑1≤i≤j≤kaibj.(2)|\lambda| = T_k+ \sum_{i=1}^k(k-i+1)a_i + \sum_{i=1}^k i b_i + \sum_{1\le i\le j\le k}a_i b_j. \tag{2}

All terms after TkT_k are nonnegative.

Fix r≥2r\ge2. If k≤r−2k\le r-2, (1) rules out local dimension at least rr. If k≥r+1k\ge r+1, (2) gives

∣λ∣≥Tr+1>Tr+1.|\lambda|\ge T_{r+1}>T_r+1.

If k=rk=r, the staircase itself has dimension r−1r-1, so dimension at least rr requires

∣λ∣≥Tr+1.|\lambda|\ge T_r+1.

Equality occurs only for b1=1b_1=1 or ar=1a_r=1, with all other excess parameters zero.

If k=r−1k=r-1, dimension at least rr requires ai,bi≥1a_i,b_i\ge1 for a common ii. Formula (2) then gives

∣λ∣−Tk≥(k−i+1)+i+1=k+2,|\lambda|-T_k \ge(k-i+1)+i+1=k+2,

hence again

∣λ∣≥Tk+k+2=Tr+1.|\lambda|\ge T_k+k+2=T_r+1.

Equality occurs exactly when ai=bi=1a_i=b_i=1 for one 1≤i≤k1\le i\le k, with all other excess parameters zero.

For k=rk=r, the two equality cases are the two outer-corner extensions

(r+1,r−1,r−2,…,1),(r,r−1,…,2,1,1)(r+1,r-1,r-2,\ldots,1), \qquad (r,r-1,\ldots,2,1,1)

of the staircase δr=(r,r−1,…,1)\delta_r=(r,r-1,\ldots,1).

For k=r−1k=r-1, the r−1r-1 equality cases ai=bi=1a_i=b_i=1 are precisely the interior-corner extensions of that same staircase. Every one of these partitions has local dimension exactly rr by (1).

Therefore the first-occurrence layer is exactly

τ(r)=r(r+1)2+1,Lr(τ(r))={δr∪{c}:c∈Add⁡(δr)},∣Lr(τ(r))∣=r+1.\boxed{ \tau(r)=\frac{r(r+1)}2+1, \qquad L_r(\tau(r)) = \left\{ \delta_r\cup\{c\}: c\in\operatorname{Add}(\delta_r) \right\}, \qquad |L_r(\tau(r))|=r+1. }

This proves the full staircase conjecture for every r≥2r\ge2.

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