General reverse-unimodality conjecture for beta-values of lists

From papers

Let $l$ be a positive integer, and let $i$ and $j$ be non-negative integers with $j>i$ . For $1\leq k\leq l$ , let $L_{i,j}^k$ be the list of length $l$ whose entries are $i$ except for $j$ in position $k$ , and define $a_{i,j}^k=\beta(L_{i,j}^k)$ . General reverse-unimodality conjecture. For fixed $i$ , $j$ , and $l$ , the sequence $\{a_{i,j}^k\}_{k=1}^l$ is reverse unimodal in $k$ . This generalizes the preceding reverse-unimodality result for the sequence $\beta(0^i,2,0^{n-i})$ ; the source gives no resolution of the conjecture.

Progress summary

Open

No publicly verified progress or resolution of this conjecture was found.

The conjecture asserts a symmetry-shaped monotonicity property for the beta-values obtained by moving one exceptional entry through a list. No published result, proof, counterexample, or named proposer was found in the retrieved sources.

Current status (as of August 2026): The conjecture remains open, with no recorded public progress or resolution.

Sources & referencesView supporting material

Primary source

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

1 week ago

The proposed reverse-unimodality statement is false already for $cd$ -monomials of degree $19$ .

Take

i=2,\qquad j=3,\qquad \ell=5,

so $i<j$ . Form the five lists containing one entry $j=3$ and four entries $i=2$ , with the position of $3$ varying from first to fifth:

\begin{aligned} L_1&=(3,2,2,2,2),\\ L_2&=(2,3,2,2,2),\\ L_3&=(2,2,3,2,2),\\ L_4&=(2,2,2,3,2),\\ L_5&=(2,2,2,2,3). \end{aligned}

All five lists have degree

3+4\cdot2+2(5-1)=19.

The exact coefficients follow from Lemmas 3.2 and 4.4 of the source. For every nonempty list $a=(a_1,\ldots,a_r)$ ,

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

with $\beta((0))=1$ . Every term on the right has degree one less, so this is a finite exact integer recurrence.

Its values are

\begin{aligned} \beta(L_1)&=20{,}985{,}240{,}240,\\ \beta(L_2)&=18{,}526{,}714{,}104,\\ \beta(L_3)&=18{,}599{,}890{,}400,\\ \beta(L_4)&=18{,}526{,}714{,}104,\\ \beta(L_5)&=20{,}985{,}240{,}240. \end{aligned}

In particular,

\beta(L_3)-\beta(L_2)=73{,}176{,}296>0.

A reverse-unimodal symmetric sequence of length five must decrease weakly from its first entry to its middle entry, requiring $\beta(L_2)\geq\beta(L_3)$ . Here the opposite strict inequality holds. Therefore the conjectured reverse unimodality fails.

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Shivam Patel · 1 day ago