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I am working on some combinatorics problems. One of my problems leads to the following question:

Is it true that the derivative of $x^n + x^{n-1} + \dots + x + 1,$ namely $nx^{n-1} + (n-1)x^{n-2} + \dots + 2x + 1$, is irreducible in $\mathbb{Z}[X]$?

I believe it is true, and I have test by computers that it is true for $n \leq 100.$

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    $\begingroup$ Note that this polynomial is nearly the same as the Type A root poset rank generating function, which was asked about at: mathoverflow.net/questions/332958/…. $\endgroup$
    – Sam Hopkins
    Commented Oct 30, 2020 at 2:05
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    $\begingroup$ Via that MO question I found matwbn.icm.edu.pl/ksiazki/aa/aa90/aa9023.pdf which you will be interested in $\endgroup$
    – Sam Hopkins
    Commented Oct 30, 2020 at 2:40
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    $\begingroup$ open problems are closer than one imagines... $\endgroup$
    – user347489
    Commented Oct 30, 2020 at 6:15
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    $\begingroup$ I wouldn't vote as duplicate (and would vote to reopen if done) since it's not so obviously covered by the linked post (which is not easily readable without familiarity to root systems), and even if it is, the remark that this is covered by the other question makes a good answer to this one. $\endgroup$
    – YCor
    Commented Oct 30, 2020 at 17:15
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    $\begingroup$ I asked this exact question on Math.SE four years ago math.stackexchange.com/q/1693589/127263 $\endgroup$
    – Wojowu
    Commented Oct 30, 2020 at 18:23

1 Answer 1

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(Turning the comments into a community wiki answer.)

This problem is discussed in Classes of polynomials having only one non-cyclotomic irreducible factor, by Borisov, Filaseta, Lam, and Trifonov (Acta Arithmetica 90 (1999), 121–153). They prove irreducibility in many special cases but the problem remains open (or at least, none of the papers that Google Scholar lists as citing this paper solves the problem).

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