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Let $P_n$ denote the poset with elements $P_n=\{1,...,n\}$ ordered by divisibility and let $L_n$ denote the distributive lattice of order ideals of $P_n$, whose elements should correspond to primitive subsequences of $\{1,...,n\}$, see https://oeis.org/A051026.

Question 1: Is $L_n$ Sperner? What is the width of $L_n$?

The width sequence start for $n \geq 2$ with 1,2,2,4,4,8,10,15,21,40,45,87 and the poset is indeed Sperner for $n \leq 15$.

Question 2: Let $F_n$ be the free distribitive lattice given as the set of order ideals of the Boolean lattice $B_n$. For which $n$ is $F_n$ Sperner? It is Sperner for $n \leq 4$. What is the sequence $a_n$ of width of $F_n$? The sequence start with 1,2,4,24 for $n \geq 1$.

More generally one can ask this question also for the distributive lattice of order ideals of the divisor lattice of $n$ (all divisors of $n$ ordered by divisibility), see the comments.

I would be very interested to see what $a_5$ and maybe $a_6$ are and see if this sequence appears in the oeis.

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    $\begingroup$ I don't know the exact classification. But even in the case when $P=m\times n$ is the product of two chains, that the distributive lattice $J(P)$ (in this case often denoted $L(m,n)$) is Sperner is a famous result of Stanley from his paper "Weyl Groups, the Hard Lefschetz Theorem, and the Sperner Property" (math.mit.edu/~rstan/pubs/pubfiles/42.pdf), though I think nowadays there are also proofs of this that avoid the Hard Lefschetz theorem. $\endgroup$
    – Sam Hopkins
    Commented Sep 29, 2023 at 12:34
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    $\begingroup$ FWIW, I just tested in Sage and $J(3 \times 3 \times 3)$ is also Sperner, so maybe my guesses are all wrong. $\endgroup$
    – Sam Hopkins
    Commented Sep 29, 2023 at 14:20
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    $\begingroup$ I can't think of any poset which is known to be Sperner but for which we don't know a "good" description of its elements. To my knowledge, no good description is known for the order ideals of $P_n$ or for a product of more than two chains (of length at least one). $\endgroup$
    – Richard Stanley
    Commented Sep 29, 2023 at 15:31
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    $\begingroup$ I know nothing about the subject, but just in case you overlooked it, I have found the sequence OEIS A341633. If it is not related to the question, I will cancel the comment. $\endgroup$
    – Fabius Wiesner
    Commented Sep 30, 2023 at 6:32
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    $\begingroup$ Dr. Larry Harper's student worked on this. Is the Free Distributive Lattice (FD(n)) Sperner? (talk by L. Harper) Tuesday, March 23, 2021 Abstract: For many years one of the most challenging problems of order theory has been to prove that FD(n) is Sperner – that the largest (middle) rank is the largest antichain. That conjecture has resisted so many good ideas that we had to consider the possibility that it is false for some large n. In this lecture we present a heuristic argument showing that it is at least asymptotically true. escholarship.org/uc/item/8wh6f7rc $\endgroup$
    – Tri
    Commented Oct 9, 2023 at 22:52

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