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The coefficients of the partition polynomials enumerate the faces of the convex, bounded polytopes called the permutahedra, or permutohedra, and the absolute value of the sum of the coefficients gives the Euler characteristic of unity for each polytope; i.e., the absolute value of the sum of each row of the array is unity. In addition, the signs of the faces alternate with dimension, and the coefficients of faces with the same dimension for each polytope have the same sign. - Tom Copeland, Nov 13 2019

The use of the term 'list partition transform' resulted from one of my first uses of these partition polynomials in relating A000262 to A084358 with their simple e.g.f.s. Other appropriate names would be the permutahedra permutohedra polynomials since they are refined Euler characteristics of the permutahedra permutohedra or the reciprocal polynomials since they give the multiplicative inverses of e.g.f.s with a constant of 1. - Tom Copeland, Oct 09 2022

M. Aguiar and F. Ardila, <a href="http://math.sfsu.edu/federico/Talks/GPatMSRI.pdf">The algebraic and combinatorial structure of generalized permutahedra [sic]</a>, MSRI Summer School July 19, 2017.

M. Aguiar and F. Ardila, <a href="https://arxiv.org/abs/1709.07504">Hopf monoids and generalized permutahedra [sic]</a>, arXiv:1709.07504 [math.CO], p. 5, 2017.

Karl-Dieter Crisman, <a href="http://www.math.gordon.edu/~kcrisman/KemenyBordaPermutahedron-final.pdf">The Borda Count, the Kemeny Rule, and the permutahedron [sic]</a>, preprint, 2014.

Karl-Dieter Crisman, <a href="http://dx.doi.org/10.1090/conm/624">The Borda Count, the Kemeny Rule, and the permutahedron [sic]</a>, in: Karl-Dieter Crisman and Michael A. Jones (eds.), The Mathematics of Decisions, Elections, and Games, Contemporary Mathematics, AMS, Vol. 624, 2014, pp. 101-134.

Indeterminate substitutions as illustrated in A356145 lead to [E] = [L][P] = [P][E]^(-1)[P] = [P][RT] and [E]^(-1) = [P][L] = [P][E][P] = [RT][P], where [E] contains the refined Eulerian partition polynomials of A145271; [E]^(-1), A356145, the inverse set to [E]; [P], the permutahedra permutohedra polynomials of this entry; [L], the classic Lagrange inversion polynomials of A134685; and [RT], the reciprocal tangent polynomials of A356144. Since [L]^2 = [P]^2 = [RT]^2 = [I], the substitutional identity, [L] = [E][P] = [P][E]^(-1) = [RT][P], [RT] = [E]^(-1)[P] = [P][L][P] = [P][E], and [P] = [L][E] = [E][RT] = [E]^(-1)[L] = [RT][E]^(-1). - Tom Copeland, Oct 05 2022

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S. Forcey, <a href="http://web.archive.org/web/20221102164731/https://www.mathsforcey.uakrongithub.eduio/~sf34/hedra.htm">The Hedra Zoo</a>

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S. Forcey, <a href="http://web.archive.org/web/20221102164731/https://www.math.uakron.edu/~sf34/hedra.htm#index">The Hedra Zoo</a>

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J. Loday, <a href="https://web.archive.org/web/20100202074906/http://www-irma.u-strasbg.fr/~loday/PAPERS/MultFAsENG2.pdf">The Multiple Facets of the Associahedron</a>

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