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A133437 Irregular triangle of coefficients C(j,k) of a partition transform for direct Lagrange inversion of an o.g.f., complementary to A134685 for an e.g.f. 15
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 (list; graph; refs; listen; history; text; internal format)
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 3-D associahedron with 14 vertices (0-D faces), 21 edges (1-D faces), 6 pentagons (2-D faces), 3 rectangles (2-D faces), 1 3-D polytope (3-D 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', or Hopf, eqn. See A001764 and A086810. - Tom Copeland, Feb 15 2014

LINKS

Table of n, a(n) for n=1..45.

F. Brown and J. Bergstrom, Inversion of series and the cohomology of the moduli spaces M_(o,n), arXiv:0910.0120 [math.AG], 2009.

T. Copeland, Lagrange a la Lah , Compositional inverse pairs, the Burgers-Hopf equation, and the Stasheff associahedra,, Generators, Inversion, and Matrix, Binomial, and Integral Transforms

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.

J. Golden, M. Paulos, M. Spradlin, and A. Volovich, Cluster polylogarithms for scattering amplitudes, arxiv: 1401.6446 (v2) [hep.th], 2014.

D. Kreimer and K. Yeats, Diffeomorphisms of quantum fields, arXiv:1610.01837 [math-ph], 2016. (pg. 7)

M. Liu, Moduli of J-holomorphic curves with lagrangian boundary conditions and open Gromov-Witten invariants for an S^1 pair, arXiv:math/0210257 [math.SG], 2002-2004.  (see Fig. 2 pg. 13)

J.-L. Loday, The Multiple Facets of the Associahedron [From Tom Copeland, Sep 29 2008]

J. McCammond, Non-crossing Partitions in Surprising Locations, American Mathematical Monthly 113 (2006) 598-610. [From Tom Copeland, Sep 29 2008]

Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arxiv: 1401.7194 [math.CO], 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)^(n-1+e(1)) * [2*(n-1)-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./(1-u.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(n-1,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)^(n-1) 1/[u_1 + u_2 * x + u_3 * x^2 + ... + u_n * x^(n-1)]^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 square-bracketed 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,.., (K-1) * bK, ..) = [(n+1)/2] * sum_{k = 2,..,n-1} PS(n-k+1,u_1=1,u_2,..,u_(n-k+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,..,(K-1)*bK,..) =  d/dt t^(n-2)*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 m-th 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+1-k}, or equivalently multiply the diagonals of A132159 by u_m. Then P(n,t) = (1, 0, 0, 0,..) [UP^(-1) * S]^(n-1) 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)

CROSSREFS

Cf. A145271, (A086810,A181289) = (reduced array, associated g(x)).

Cf.  A001764, A007318, A033282, A086810, A111785, A126216, A129185, A132159, A133314, A134685, A139605.

Sequence in context: A035877 A086494 A107414 * A245692 A182126 A266511

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

STATUS

approved

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Last modified December 10 04:09 EST 2016. Contains 278993 sequences.