

A133437


Irregular triangle of coefficients of a partition transform for direct Lagrange inversion of an o.g.f., complementary to A134685 for an e.g.f.; Normalized by the factorials, these are signed, refined face polynomials of the associahedra


23



1, 2, 12, 6, 120, 120, 24, 1680, 2520, 720, 360, 120, 30240, 60480, 20160, 20160, 5040, 5040, 720, 665280, 1663200, 907200, 604800, 362880, 60480, 181440, 40320, 40320, 20160, 5040, 17297280, 51891840, 39916800, 19958400, 6652800, 19958400, 6652800, 1814400, 3628800, 1814400, 1814400, 362880, 362880, 362880, 40320
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OFFSET

1,2


COMMENTS

Let f(t) = u(t)  u(0) = Sum_{n>=1} u_n * t^n.
If u_1 is not equal to 0, then the compositional inverse for f(t) is given by g(t) = Sum_{j>=1) P(n,t) where, with u_n denoted by (n'),
P(1,t) = (1')^(1) * [ 1 ] * t
P(2,t) = (1')^(3) * [ 2 (2') ] * t^2 / 2!
P(3,t) = (1')^(5) * [ 12 (2')^2  6 (1')(3') ] * t^3 / 3!
P(4,t) = (1')^(7) * [ 120 (2')^3 + 120 (1')(2')(3')  24 (1')^2 (4') ] * t^4 / 4!
P(5,t) = (1')^(9) * [ 1680 (2')^4  2520 (1') (2')^2 (3') + 720 (1')^2 (2') (4') + 360 (1')^2 (3')^2  120 (1')^3 (5') ] * t^5 / 5!
P(6,t) = (1')^(11) * [ 30240 (2')^5 + 60480 (1') (2')^3 (3')  20160 (1')^2 (2') (3')^2  20160 (1')^2 (2')^2 (4') + 5040 (1')^3 (2')(5') + 5040 (1')^3 (3')(4')  720 (1')^4 (6') ] * t^6 / 6!
P(7,t) = (1')^(13) * [ 665280 (2')^6  1663200 (1')(2')^4(3') + (1')^2 [907200 (2')^2(3')^2 + 604800 (2')^3(4')]  (1')^3 [362880 (2')(3')(4') + 60480 (3')^3 + 181440 (2')^2(5')] + (1')^4 [40320 (2')(6') + 40320 (3')(5') + 20160 (4')^2]  5040 (1')^5(7')] * t^7 / 7!
P(8,t) = (1')^(15) * [ 17297280 (2')^7 + 51891840 (1')(2')^5(3')  (1')^2 [39916800 (2')^3(3')^2 + 19958400 (2')^4(4')] + (1')^3 [6652800 (2')(3')^3 + 19958400 (2')^2(3')(4') + 6652800 (2')^3(5')]  (1')^4 [1814400 (2')(4')^2 + 3628800 (2')(3')(5') + 1814400 (2')^2(6') + 1814400 (3')^2(4')] + (1')^5 [362880 (2')(7') + 362880 (3')(6') + 362880 (4')(5')]  40320 (1')^6(8')] * t^8 / 8!
...
See A134685 for more information.
A111785 is obtained from A133437 by dividing through the bracketed terms of the P(n,t) by n! and unsigned A111785 is a refinement of A033282 and A126216.  Tom Copeland, Sep 28 2008
For relation to the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see the Loday and McCammond links. E.g., P(5,t) = (1')^(9) * [ 14 (2')^4  21 (1') (2')^2 (3') + 6 (1')^2 (2') (4')+ 3 (1')^2 (3')^2  1 (1')^3 (5') ] * t^5 is related to the 3D associahedron with 14 vertices (0D faces), 21 edges (1D faces), 6 pentagons (2D faces), 3 rectangles (2D faces), 1 3D polytope (3D faces). Summing over faces of the same dimension gives A033282 or A126216.  Tom Copeland, Sep 29 2008
A relation between this Lagrange inversion for an o.g.f. and partition polynomials formed from the "refined Lah numbers" A130561 is presented in the link "Lagrange a la Lah" along with umbral binary tree representations.
With f(x,t) = x + t*Sum_{n>=2} u_n*x^n, the compositional inverse in x is related to the velocity profile of particles governed by the inviscid Burgers's, or Hopf, eqn. See A001764 and A086810.  Tom Copeland, Feb 15 2014
Newton was aware of this power series expansion for series reversion. See the Ferraro reference p. 75 eqn. 52.  Tom Copeland, Jan 22 2017
The coefficients of the partition polynomials divided by the associated factorial enumerate the faces of the convex, bounded polytopes called the associahedra, and the absolute value of the sum of the renormalized 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 either n! (unnormalized) or unity (normalized). 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
With u_1 = 1 and the other u_n replaced by suitably signed partition polynomials of A263633, the partition polynomials enumerating the refined noncrossing partitions of A134264 with a shift in indices are obtained (cf. In the Realm of Shadows).  Tom Copeland, Nov 16 2019
Relations between associahedra and oriented nsimplices are presented by Halvorson and by Street.  Tom Copeland, Dec 08 2019
Let f(x;t,n) = x  t * x^(n+1), giving u_1 = (1') = 1 and u_(n+1) = (n+1) = t. Then inverting in x with t a parameter gives finv(x;t,n) = Sum_{j>=0} {binomial((n+1)*j,j) / (n*j + 1)} * t^j * x^(n*j + 1), which gives the Catalan numbers for n=1, and the FussCatalan sequences for n>1 (see A001764, n=2). Comparing this with the same result in A134264 gives relations between the faces of associahedra and noncrossing partitions (and other combinatorial constructs related to these inversion formulas and those listed in A145271).  Tom Copeland, Jan 27 2020
From Tom Copeland, Mar 24 2020: (Start)
There is a mapping between the faces of K_n, the associahedron of dimension (n1), and polygon dissections. The dissecting noncrossing diagonals (i.e., nonintersecting in the interior) form subpolygons. Assign the indeterminate x_k to a subpolygon where k = (number of vertices of the subpolygon)  1. Multiply the x_k together to form the monomials for the inversion formula.
For the 3dimensional associahedron K_4, the fundamental polygon is the hexagon, which can be dissected into pentagons, associated to x_4; tetragons , to x_3; and triangles, to x_2; for example, there are six distinguished partitions of the hexagon into one triangle and one pentagon, sharing two vertices, associated to the monomial 6 * x_2 * x_4 since the unshared vertex of the triangle can be moved consecutively from one vertex of the heaxagon to the next. This term corresponds to 720 (1')^2 (2') (4') / 5! in P(5,t) above, denumerating the six pentagonal facets of K_4. (End)


REFERENCES

G. Ferraro, The Rise and Development of the Theory of Series up to the Early 1820s, Springer Science and Business Media, 2007.
H. Halvorson (editor), Deep Beauty: Understanding the Quantum World Through Innovation, Cambridge Univ. Press, 2011.


LINKS

Table of n, a(n) for n=1..45.
M. Aguiar and F. Ardila, The algebraic and combinatorial structure of generalized permutahedra, MSRI Summer School July 19, 2017.
M. Aguiar and F. Ardila, Hopf monoids and generalized permutahedra, arXiv:1709.07504 [math.CO], p. 6, 2017.
N. ArkaniHamed, Y. Bai, S. He, and G. Yan, Scattering forms and the positive geometry of kinematics, color, and the worldsheet , arXiv:1711.09102 [hepth], 2017.
D. Armstrong, Catalan numbers: From EGS to BEG, MIT Combinatorics Seminar, May 15, 2015.
P. Balduf, The propagator and diffeomorphisms of an interacting field theory, Master's thesis, submitted to the Institut für Physik, MathematischNaturwissenschaftliche Fakultät, HumboldtUniverstität, Berlin, 2018, p. 32.
P. Balduf, Perturbation theory of transformed quantum fields, arXiv:1905.00686 [mathph], 2019, (see Example 2.3 on p. 4 and Lemma 2.7 on p. 5).
L. Berry, S. Forcey, M. Ronco, and P. Showers, Polytopes and Hopf algebras of painted trees: Fan graphs and Stellohedra, arXiv:1608.08546 [math.CO], 2018.
TaiDanae Bradley, Associahedra: The Shapes of Multiplication, an episode on YouTube of PBS Infinite Series, Nov 2017.
TaiDanae Bradley, What is an Operad? Part 2, a post on the blog Math3ma, Oct 2017.
F. Brown and J. Bergstrom, Inversion of series and the cohomology of the moduli spaces M_(o,n), arXiv:0910.0120 [math.AG], 2009.
V. Buchstaber, Toric Topology of Stasheff polytopes, Manchester Institute for Mathematical Sciences Eprint 2007.232, School of Mathematics, University of Manchester, 2007.
J. Carter, A. Crans, E. Elhamdadi, E. Karadayi, and M. Saito, Cohomology of Frobenius Algebras and the YangBaxter Equation, arXiv:0801.2567 [math.QA], 2008.
B. Casselman, Strange Associations, AMS Feature Column, Nov 2007.
X. Chen, T. Yi, F. He, Z. He, and Z. Dong, An Improved Generalized Chirp Scaling Algorithm Based on Lagrange Inversion Theorem for HighResolution Low Frequency Synthetic Aperture Radar Imaging, Remote Sensing, Vol. 11, Issue 16, 2019.
Tom Copeland, Lagrange a la Lah, 2011.
Tom Copeland Compositional inverse pairs, the BurgersHopf equation, and the Stasheff associahedra, 2014.
Tom Copeland, Compositional inversion and generating functions in algebraic geometry, MathOverflow question, 2014.
Tom Copeland, Guises of the Stasheff polytopes, associahedra for the Coxeter A_n root system?, MathOverflow question, 2014.
Tom Copeland, An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes, MathOverflow question, 2014.
Tom Copeland, Why is there a connection between enumerative geometry and nonlinear waves?, MathOverflow answer, 2014.
Tom Copeland, Important formulas in combinatorics, MathOverflow answer, 2015.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.
Tom Copeland, Formal group laws and binomial Sheffer sequences, 2018.
Tom Copeland, In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms, 2019.
L. Cruz, A. Kniss, and S. Weinzierl, Properties of scattering forms and their relation to associahedra, arXiv:1711.07942 [hepth], 2018.
S. Devadoss, B. Fehrman, T. Heath, and A. Vashist, Moduli spaces of punctured Poincaré discs, arXiv:1109.2830 [math.AT], 2011.
S. Devadoss, T. Heath, and W. Vipismakul, Deformation of bordered surfaces and convex polytopes, Notices of the AMS, April 2011, Volume 58, Issue 04.
C. Dupont and B. Vallette, Browns' moduli spaces of curves and the gravity operad, arXiv:1590.08840 [math.AG], p. 15, 2015.
H. Einziger, Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions, The George Washington University, ProQuest Dissertations Publishing, 2010. 3417955. (Cf. p. 61.)
H. Figueroa and J. GraciaBondia, Combinatorial Hopf algebras in quantum field theory I, arXiv:0408145 [hepth], 2005, (p. 44).
S. Forcey, The Hedra Zoo
A. Frabetti and D. Manchon, Five interpretations of Fa`a di Bruno’s formula , arXiv:1402.5551 [math.CO], 2014, p. 11.
X. Gao, S. He, and Y. Zhan, Labelled tree graphs, Feynman diagrams and disk integrals , arXiv:1708.08701 [hepth], 2017.
E. Ghys, A Singular Mathematical Promenade, arXiv:1612.06373 [math.GT], 2017.
J. Golden, M. Paulos, M. Spradlin, and A. Volovich, Cluster polylogarithms for scattering amplitudes, arxiv: 1401.6446 (v2) [hep.th], 2014.
A. Hanson and J. Sha, A Contour Integral Representation for the Dual FivePoint Function and a Symmetry of the Genus Four Surface in R6, arXiv preprint arXiv:0510064 [mathph], 2005.
S. He, Scattering from Geometries, presentation at the Interdisciplinary Center for Theoretical Study at the University of Science and Technology of China, 2017.
D. Herceg, L. Petkovic, and M. Petkovic, On Schröder’s families of rootfinding methods, Journal of Computational and Applied Mathematics, Vol. 233, Issue 8, 15 February 2010, Pages 17551762, (see p. 1760, below eqn. 19).
D. Jackson, A. Kempf, and A. Morales, A robust generalization of the Legendre transform for QFT, arXiv:1612.0046 [hepth], 2017.
A. Kirillov, On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, FussCatalan, Universal Tutte and Reduced Polynomials, arXiv preprint arXiv:1502.00426 [math.RT], 2016, p. 150.
D. Kreimer and K. Yeats, Diffeomorphisms of quantum fields, arXiv:1610.01837 [mathph], 2016 (p. 7).
C. Lenart, Lagrange inversion and Schur functions, Journal of Algebraic Combinatorics, Vol. 11, Issue 1, pp. 6978, 2000, (see p. 70, Eqn. 1.2).
M. Lin, Graph Cohomology, 2016.
M. Liu, Moduli of Jholomorphic curves with lagrangian boundary conditions and open GromovWitten invariants for an S^1 pair, arXiv:math/0210257 [math.SG], 20022004. (See Fig. 2, p. 13.)
JL. Loday, Realization of the Stasheff polytope, arXiv:0212126v6 [math.AT], 2002.
JL. Loday, The Multiple Facets of the Associahedron
JL. Loday and B. Vallette, Algebraic Operads, version 0.99, p. 442, 2012.
A. Mahmoud and K. Yeats, Diffeomorphisms of Scalar Quantum Fields via Generating Functions, arXiv:2007.12341 [mathph], 2021, (see Lemma 8.2 on p. 15).
MathOverflow, Analogue of conic sections for the permutohedra, associahedra, and noncrossing partitions, an MO question posed by T. Copeland, 2017.
MathOverflow, Loday's characterization and enumeration of faces of associahedra (Stasheff polytopes), an MO question posed by T. Copeland and answered by R. Davis, 2017.
J. McCammond, Noncrossing Partitions in Surprising Locations, American Mathematical Monthly 113 (2006) 598610.
S. Mizera, Combinatorics and Topology of KawaiLewellenTye Relations, arXiv:1706.08527 [hepth], 2017.
V. Pilaud, The Associahedron and its Friends, presentation for Séminaire Lotharingien de Combinatoire, April 4  6, 2016.
J. Pitman and R. Stanley, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, arXiv:math/9908029 [math.CO], 1999.
Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
J. Stasheff, What is ... an Operad?, Notices of the American Mathematical Society 51 (6), JuneJuly 2004, 630631.
R. Street, The algebra of oriented simplexes, Journal of Pure and Applied Algebra, 49, pp. 283335, 1987.
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 101.
B. Vallette, Algebra + Homotopy = Operad, arXiv:1202.3245 [math.AT], 2012 (p. 21).
J. Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.


FORMULA

The bracketed partitions of P(n,t) are of the form (u_1)^e(1) (u_2)^e(2) ... (u_n)^e(n) with coefficients given by (1)^(n1+e(1)) * [2*(n1)e(1)]! / [ (e(2))! * (e(3))! * ... * (e(n))! ].
From Tom Copeland, Sep 06 2011: (Start)
Let h(t) = 1/(df(t)/dt)
= 1/Ev[u./(1u.t)^2]
= 1/((u_1) + 2*(u_2)*t + 3*(u_3)*t^2 + 4*(u_4)*t^3 + ...),
where Ev denotes umbral evaluation.
Then for the partition polynomials of A133437,
n!*P(n,t) = ((t*h(y)*d/dy)^n) y evaluated at y=0,
and the compositional inverse of f(t) is
g(t) = exp(t*h(y)*d/dy) y evaluated at y=0.
Also, dg(t)/dt = h(g(t)). (End)
From Tom Copeland, Oct 20 2011: (Start)
With exp[x* PS(.,t)] = exp[t*g(x)] = exp[x*h(y)d/dy] exp(t*y) eval. at y=0, the raising/creation and lowering/annihilation operators defined by R PS(n,t)=PS(n+1,t) and L PS(n,t) = n*PS(n1,t) are
R = t*h(d/dt) = t* 1/[(u_1) + 2*(u_2)*d/dt + 3*(u_3)*(d/dt)^2 + ...] and
L = f(d/dt) = (u_1)*d/dt + (u_2)*(d/dt)^2 + (u_3)*(d/dt)^3 + ....
Then P(n,t) = (t^n/n!) dPS(n,z)/dz eval. at z=0. (Cf. A139605, A145271, and link therein to Mathemagical Forests for relation to planted trees on p. 13.) (End)
The bracketed partition polynomials of P(n,t) are also given by (d/dx)^(n1) 1/[u_1 + u_2 * x + u_3 * x^2 + ... + u_n * x^(n1)]^n evaluated at x=0.  Tom Copeland, Jul 07 2015
From Tom Copeland, Sep 20 2016: (Start)
Let PS(n,u1,u2,...,un) = P(n,t) / t^n, i.e., the squarebracketed part of the partition polynomials in the expansion for the inverse in the comment section, with u_k = uk.
Also let PS(n,u1=1,u2,...,un) = PB(n,b1,b2,...,bK,...) where each bK represents the partitions of PS, with u1 = 1, that have K components or blocks, e.g., PS(5,1,u2,...,u5) = PB(5,b1,b2,b3,b4) = b1 + b2 + b3 + b4 with b1 = u5, b2 = 6 u2 u4 + 3 u3^2, b3 = 21 u2^2 u3, and b4 = 14 u2^4.
The relation between solutions of the inviscid Burgers' equation and compositional inverse pairs (cf. A086810) implies that, for n > 2, PB(n, 0 * b1, 1 * b2, ..., (K1) * bK, ...) = [(n+1)/2] * Sum_{k = 2..n1} PS(nk+1,u_1=1,u_2,...,u_(nk+1)) * PS(k,u_1=1,u_2,...,u_k).
For example, PB(5,0 * b1, 1 * b2, 2 * b3, 3 * b4) = 3 * 14 u2^4  2 * 21 u2^2 u3 + 1 * 6 u2 u4 + 1 * 3 u3^2  0 * u5 = 42 u2^4  42 u2^2 u3 + 6 u2 u4 + 3 u3^2 = 3 * [2 * PS(2,1,u2) * PS(4,1,u2,...,u4) + PS(3,1,u2,u3)^2] = 3 * [ 2 * (u2) (5 u2^3 + 5 u2 u3  u4) + (2 u2^2  u3)^2].
Also, PB(n,0*b1,1*b2,...,(K1)*bK,...) = d/dt t^(n2)*PS(n,u1=1/t,u2,...,un)_{t=1} = d/dt (1/t)*PS(n,u1=1,t*u2,...,t*un)_{t=1}.
(End)
From Tom Copeland, Sep 22 2016: (Start)
Equivalent matrix computation: Multiply the mth diagonal (with m=1 the index of the main diagonal) of the lower triangular Pascal matrix A007318 by f_m = m!*u_m = (d/dx)^m f(x) evaluated at x=0 to obtain the matrix UP with UP(n,k) = binomial(n,k) f_{n+1k}, or equivalently multiply the diagonals of A132159 by u_m. Then P(n,t) = (1, 0, 0, 0, ...) [UP^(1) * S]^(n1) FC * t^n/n!, where S is the shift matrix A129185, representing differentiation in the basis x^n//n!, and FC is the first column of UP^(1), the inverse matrix of UP. These results follow from A145271 and A133314.
Also, P(n,t) = (1, 0, 0, 0, ...) [UP^(1) * S]^n (0, 1, 0, ...)^T * t^n/n! in agreement with A139605. (End)
A recursion relation for computing each partition polynomial of this entry from the lower order polynomials and the coefficients of the refined Lah polynomials of A130561 is presented in the blog entry "Formal group laws and binomial Sheffer sequences."  Tom Copeland, Feb 06 2018
The derivative of the partition polynomials of A350499 with respect to a distinguished indeterminate give polynomials proportional to those of this entry. The connection of this derivative relation to the inviscid BurgersHopf evolution equation is given in a reference for that entry.  Tom Copeland, Feb 19 2022


CROSSREFS

Cf. A145271, (A086810,A181289) = (reduced array, associated g(x)).
Cf. A001764, A007318, A033282, A086810, A111785, A126216, A129185.
Cf. A130561, A132159, A133314, A134264, A134685, A139605, A263633.
Sequence in context: A086494 A354484 A107414 * A245692 A182126 A334143
Adjacent sequences: A133434 A133435 A133436 * A133438 A133439 A133440


KEYWORD

sign,tabf


AUTHOR

Tom Copeland, Jan 27 2008


EXTENSIONS

Missing coefficient in P(6,t) replaced by Tom Copeland, Nov 06 2008
P(7,t) and P(8,t) data added by Tom Copeland, Jan 14 2016
Title modified by Tom Copeland, Jan 13 2020


STATUS

approved



