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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.. 11
1, -2, 12, -6, -120, 120, -24, 1680, -2520, 720, 360, -120, -30240, 60480, -20160, -20160, 5040, 5040, -720 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Let f(t) = u(t) - u(0) = sum(n=1,2,...) 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!

...

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..19.

F. Brown and J. Bergstrom, Inversion of series and the cohomology of the moduli spaces M_(o,n)

Tom Copeland, Compositional inverse pairs, the Burgers-Hopf equation, and the Stasheff associahedra,

T. Copeland, Lagrange a la Lah

S. Devadoss, B. Fehrman, T. Heath, and A. Vashist, Moduli spaces of punctured Poincare discs

S. Devadoss, T. Heath, and W. Vipismakul, Deformation of bordered surfaces and convex polytopes

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

M. Liu, Moduli of J-holomorphic curves with lagrangian boundary conditions and open Gromov-Witten invariants for an S^1 pair  (see Fig. 2 pg. 13)

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

J. McCammond, Non-crossing Partitions in Surprising Locations [From Tom Copeland, Sep 29 2008]

Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arxiv: 1401.7194 [math.CO]

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)

CROSSREFS

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

Sequence in context: A035877 A086494 A107414 * A245692 A182126 A014964

Adjacent sequences:  A133434 A133435 A133436 * A133438 A133439 A133440

KEYWORD

sign,tabf,more,changed

AUTHOR

Tom Copeland, Jan 27 2008

EXTENSIONS

Replaced a missing coefficient in P(6,t). - Tom Copeland, Nov 06 2008

STATUS

approved

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Last modified October 20 00:14 EDT 2014. Contains 248320 sequences.