Let beta(L) denote the beta-value associated with a list L, and let list concatenation be written by juxtaposition. Let m,n be non-negative integers with n>m, and let L and M be lists. Skewed-pair inequality conjecture. If M is non-empty, then beta(M,m,L,n)≥beta(M,n,L,m). The conjecture is motivated by the preceding theorem and proposes that, in this placement, exchanging the more separated pair for the more balanced ordering cannot decrease the beta-value; the source gives no resolution.

Skewed-pair inequality conjecture for the cd-index

From papers

Let $\beta(L)$ denote the beta-value associated with a list $L$ , and let list concatenation be written by juxtaposition. Let $m,n$ be non-negative integers with $n>m$ , and let $L$ and $M$ be lists. Skewed-pair inequality conjecture. If $M$ is non-empty, then

\beta(M,m,L,n)\geq\beta(M,n,L,m).

The conjecture is motivated by the preceding theorem and proposes that, in this placement, exchanging the more separated pair for the more balanced ordering cannot decrease the beta-value; the source gives no resolution.

Progress summary

Open

The conjecture remains unresolved: the scan found only related work, not a proof or counterexample.

The conjecture asks whether, for nonempty $M$ and $n>m$ , exchanging the two marked entries in $\beta(M,m,L,n)$ cannot decrease the value. Mahajan’s 2002 work studies related $\beta$ -coefficient inequalities for the Boolean lattice’s $cd$ -index, but the retrieved material does not identify this exact conjecture or resolve it.

Current status (as of August 2026): The skewed-pair inequality remains open; no verified proof, disproof, counterexample, or claimed resolution was found in the retrieved sources.

Sources

export.arxiv.org

Sources & referencesView supporting material

Primary source

Swapneel Mahajan, “The cd-index of the Boolean lattice”, arXiv:math/0211390 (2002).

1 week ago

The conjecture is false for $cd$ -monomials of degree $35$ .

Let $\beta(a_1,\ldots,a_r)$ denote the coefficient of

c^{a_1}dc^{a_2}d\cdots dc^{a_r}

in the Boolean $cd$ -index. Choose

M=(9),\qquad L=(13,0),\qquad m=2,\qquad n=3.

Then $M$ is nonempty and $n>m$ , satisfying every hypothesis.

The coefficient recurrence established by Lemmas 3.2 and 4.4 is

\begin{aligned} \beta(a_1,\ldots,a_r) ={}&\sum_{\substack{1\leq s\leq r\a_s>0}} \beta(a_1,\ldots,a_s-1,\ldots,a_r)\\ &+\sum_{s=1}^{r-1} \beta(a_1,\ldots,a_{s-1},a_s+a_{s+1}+1, a_{s+2},\ldots,a_r), \end{aligned}

with $\beta((0))=1$ . Every term on the right has degree one smaller, so this determines each coefficient by a finite exact integer computation.

Applying the recurrence gives

\begin{aligned} \beta(M,m,L,n) &=\beta(9,2,13,0,3) =257{,}721{,}391{,}969{,}561{,}080,\\ \beta(M,n,L,m) &=\beta(9,3,13,0,2) =257{,}768{,}402{,}719{,}464{,}000. \end{aligned}

Both lists have degree

9+2+13+0+3+2(5-1)=35.

Nevertheless,

\beta(M,m,L,n)-\beta(M,n,L,m) =-47{,}010{,}749{,}902{,}920<0.

Thus the asserted weak inequality fails strictly, disproving the conjecture.

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Shivam Patel · 1 day ago