

A074060


Graded dimension of the cohomology ring of the moduli space of npointed curves of genus 0 satisfying the associativity equations of physics (also known as the WDVV equations).


5



1, 1, 1, 1, 5, 1, 1, 16, 16, 1, 1, 42, 127, 42, 1, 1, 99, 715, 715, 99, 1, 1, 219, 3292, 7723, 3292, 219, 1, 1, 466, 13333, 63173, 63173, 13333, 466, 1, 1, 968, 49556, 429594, 861235, 429594, 49556, 968, 1, 1, 1981, 173570, 2567940, 9300303, 9300303, 2567940, 173570, 1981, 1
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OFFSET

3,5


COMMENTS

Combinatorial interpretations of Lagrange inversion (A134685) and the 2Stirling numbers of the first kind (A049444 and A143491) provide a combinatorial construction for A074060 (see first Copeland link). For relations of A074060 to other arrays see second Copeland link page 19.  Tom Copeland, Sep 28 2008
These Poincare polynomials for the compactified moduli space of rational curves are presented on p. 5 of Lando and Zvonkin as well as those for the noncompactified Poincare polynomials of A049444 in factorial form.  Tom Copeland, Jun 13 2021


LINKS

Table of n, a(n) for n=3..57.
Tom Copeland, Combinatorics of OEISA074060, Posted Sept. 2008.
Tom Copeland, Mathemagical Forests v2, Posted June 2008.
S. Keel, Intersection theory of moduli space of stable npointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545574.
M. Kontsevich and Y. Manin, Quantum cohomology of a product, (with Appendix by R. Kaufmann), Inv. Math. 124, f. 13 (1996) 313339.
M. Kontsevich and Y. Manin, Quantum cohomology of a product, arXiv:qalg/9502009, 1995.
S. Lando and A. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, 141, Springer, 2004.
Y. Manin, Generating functions in algebraic geometry and sums over trees, arXiv:alggeom/9407005, 1994.  Tom Copeland, Dec 10 2011
M. A. Readdy, The preWDVV ring of physics and its topology, preprint, 2002.


FORMULA

Define offset to be 0 and P(n,t) = (1)^n Sum_{j=0..n2} a(n2,j)*t^j with P(1,t) = 1 and P(0,t) = 1, then H(x,t) = 1 + exp(P(.,t)*x) is the compositional inverse in x about 0 of G(x,t) in A049444. H(x,0) = exp(x)  1, H(x,1) = 1 + exp( 2 + W( exp(2) * (2x) ) ) and H(x,2) = 1  (1+2*x)^(1/2), where W is a branch of the Lambert function such that W(2*exp(2)) = 2.  Tom Copeland, Feb 17 2008
Let offset=0 and g(x,t) = (1t)/((1+x)^(t1)t), then the nth row polynomial of the table is given by [(g(x,t)*D_x)^(n+1)]x with the derivative evaluated at x=0.  Tom Copeland, Jun 01 2008
With the notation in Copeland's comments, dH(x,t)/dx = g(H(x,t),t).  Tom Copeland, Sep 01 2011
The term linear in x of [x*g(d/dx,t)]^n 1 gives the nth row polynomial with offset 1. (See A134685.)  Tom Copeland, Oct 21 2011


EXAMPLE

Viewed as a triangular array, the values are
1;
1, 1;
1, 5, 1;
1, 16, 16, 1;
1, 42, 127, 42, 1; ...


MAPLE

DA:=((1+t)*A(u, t)+u)/(1t*A(u, t)): F:=0: for k from 1 to 10 do F:=map(simplify, int(series(subs(A(u, t)=F, DA), u, k), u)); od: # Eric Rains, Apr 02 2005


MATHEMATICA

DA = ((1+t) A[u, t] + u)/(1  t A[u, t]); F = 0;
Do[F = Integrate[Series[DA /. A[u, t] > F, {u, 0, k}], u], {k, 1, 10}];
(cc = CoefficientList[#, t]; cc Denominator[cc[[1]]])& /@ Drop[ CoefficientList[F, u], 2] // Flatten (* JeanFrançois Alcover, Oct 15 2019, after Eric Rains *)


CROSSREFS

Cf. A074059. 2nd diagonal is A002662.
Sequence in context: A156920 A174044 A174159 * A157637 A157181 A347974
Adjacent sequences: A074057 A074058 A074059 * A074061 A074062 A074063


KEYWORD

nonn,tabl


AUTHOR

Margaret A. Readdy, Aug 16 2002


EXTENSIONS

More terms from Eric Rains, Apr 02 2005


STATUS

approved



