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The relation between the refined face numbers of the permutohedra and the formal series expansion of the reciprocal of a function (exponential generating function, formal Taylor series) is given in the OEIS entries A049019 and A133314 and is reiterated in this MO-Q. What are some early references for this relation?

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Tom Copeland
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    Could you reproduce the statement please? – მამუკა ჯიბლაძე May 06 '16 at 05:58
  • Sorry not for me. Maybe you could state something in the question? The OEIS links both start labyrinths of links where I became lost very soon. The last one does not contain any mention of permutohedra itself. The first one does, but I could not extract a clean statement from there. For example, what does "refined face polynomial" mean? – მამუკა ჯიბლაძე May 06 '16 at 08:37
  • What I am familiar with is the associahedral case. Do you mean that associahedra are to ogf as permutohedra to egf? – მამუკა ჯიბლაძე May 06 '16 at 08:43
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    Thanks, this comment was indeed helpful. I now understand. Still I believe you make it hard for a first reader. Following the link one sees "This array is related to the reciprocal of an e.g.f. as sketched in A133314." and then an illustration, which makes it clear in principle, but still does not provide any rigorous statement. What is still not clear to me for example, is whether it is a proven fact or a conjectural relationship? – მამუკა ჯიბლაძე May 06 '16 at 09:01
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    You should certainly reproduce the statement in the body of the question. – Neil Strickland May 06 '16 at 09:27
  • Thanks for the link, but note that for new readers not already thoroughly familiar with the ideas the only way to realize that they must not read this question but instead turn to another one is to scroll all the way down and read two of your last comments. – მამუკა ჯიბლაძე May 06 '16 at 19:58
  • Also I cannot help noticing that both of the questions you linked to as examples of low cost benefit ratio are full of complaints similar to those here from various users, including the main contributors to this ratio. – მამუკა ჯიბლაძე May 06 '16 at 20:01
  • @TomCopeland I'm complaining because in all (another typical hyperbole) other cases here on MO I need much less effort to figure out what exactly is the question about. I'm used to it. – მამუკა ჯიბლაძე May 06 '16 at 22:26
  • Geez, ANYONE who had already come across the earliest notes of the connection between permutohedra and multiplicative inversion would have known exactly what I was talking about. – Tom Copeland Oct 24 '21 at 04:02
  • The question was addressed to that audience, not as an introduction nor motivation for the uninitiated, particularly the rude and those lacking the self-motivation to absorb the simple presentations in the OEIS entry. My experience has been that those who strongly complained about other questions have contributed absolutely zero even after refinement and elaboration on the Q (this applies just as well to my highly upvoted, favorited posts with multiple answers). – Tom Copeland Oct 24 '21 at 04:02
  • I suspect the only reason the MO-Q https://mathoverflow.net/questions/144790/multicatalan-numbers was answered was that Gessel and I were already familiar with partition polynomials and their related multinomial coefficients related to compositional inversion and could translate the multinomial expression there into a more familiar form. I have never ever seen the multinomial expressed that way before or since. – Tom Copeland Mar 02 '22 at 06:27

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