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

At least 23 years old · documented by 2002

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.

References

Primary source

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

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reader-supplied calculation claims to disprove the conjecture in degree $35$ , but the claim has not been independently verified.

The conjecture asserts that, for nonempty $M$ and $n>m$ , swapping the two marked entries cannot decrease the beta-value. The stored source records no resolution.

Posted attempt

A reader-supplied exact recurrence calculation claims a strict counterexample with $M=(9)$ , $L=(13,0)$ , $m=2$ , and $n=3$ : $\beta(9,2,13,0,3)=257{,}721{,}391{,}969{,}561{,}080$ is claimed to be less than $\beta(9,3,13,0,2)=257{,}768{,}402{,}719{,}464{,}000$ . Both lists are claimed to have degree $35$ , so this would completely disprove the conjecture. The attempt has not been independently verified.

Current status (as of August 2026): The conjecture has an unverified claimed counterexample in degree $35$ ; absent independent verification, its mathematical status remains unsettled.

Sources

export.arxiv.org

CounterexampleThis solution needs a summarySee full solutionHide full solution

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.