Margaret A. Readdy - cp's OEIS frontend

User: Margaret A. Readdy

Margaret A. Readdy has authored 3 sequences.

A256161 Triangle of allowable Stirling numbers of the second kind a(n,k).

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 11, 6, 1, 1, 5, 26, 23, 9, 1, 1, 6, 57, 72, 50, 12, 1, 1, 7, 120, 201, 222, 86, 16, 1, 1, 8, 247, 522, 867, 480, 150, 20, 1, 1, 9, 502, 1291, 3123, 2307, 1080, 230, 25, 1, 1, 10, 1013, 3084, 10660, 10044, 6627, 2000, 355, 30, 1

Offset: 1

Margaret A. Readdy, Mar 16 2015

Row sums = A007476 starting (1, 2, 4, 9, 23, 65, 199, 654, 2296, 8569, ...).

a(n,k) counts restricted growth words of length n in the letters {1, ..., k} where every even entry appears exactly once.

			a(4,1) = 1 via 1111;
a(4,2) = 3 via 1211, 1121, 1112;
a(4,3) = 4 via 1213, 1231, 1233, 1123;
a(4,4) = 1 via 1234.
Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  4,  1;
  1,  4, 11,  6,  1;
  ...

Yue Cai and Margaret Readdy, Negative q-Stirling numbers, arXiv:1506.03249 [math.CO], 2015.

Cf. A007476 (row sums), A246118 (essentially the same triangle).

a[, 1] = a[n, n_] = 1;
a[n_, k_] := a[n, k] = a[n-1, k-1] + Ceiling[k/2] a[n-1, k];
Table[a[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 15 2018 *)

a(n,k) = a(n-1,k-1) + ceiling(k/2)*a(n-1,k) for n >= 1 and 1 <= k <= n with boundary conditions a(n,0) = KroneckerDelta[n,0].
a(n,2) = n-1.
a(n,n-1) = floor(n/2)*ceiling(n/2).

A074060 Graded dimension of the cohomology ring of the moduli space of n-pointed stable curves of genus 0 satisfying the associativity equations of physics (also known as the WDVV equations).

Original entry on oeis.org

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

Offset: 3

Margaret A. Readdy, Aug 16 2002

Combinatorial interpretations of Lagrange inversion (A134685) and the 2-Stirling 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 non-compactified Poincare polynomials of A049444 in factorial form. - Tom Copeland, Jun 13 2021

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

Tom Copeland, Combinatorics of OEIS-A074060, Posted Sept. 2008.
Tom Copeland, Mathemagical Forests v2, Posted June 2008.
S. Keel, Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545-574.
M. Kontsevich and Y. Manin, Quantum cohomology of a product, (with Appendix by R. Kaufmann), Inv. Math. 124, f. 1-3 (1996) 313-339.
M. Kontsevich and Y. Manin, Quantum cohomology of a product, arXiv:q-alg/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:alg-geom/9407005, 1994. - _Tom Copeland_, Dec 10 2011
M. A. Readdy, The pre-WDVV ring of physics and its topology, preprint, 2002.

Cf. A074059. 2nd diagonal is A002662.

DA:=((1+t)*A(u,t)+u)/(1-t*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

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 (* Jean-François Alcover, Oct 15 2019, after Eric Rains *)

Define offset to be 0 and P(n,t) = (-1)^n Sum_{j=0..n-2} a(n-2,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) * (2-x) ) ) 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) = (1-t)/((1+x)^(t-1)-t), then the n-th 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 n-th row polynomial with offset 1. (See A134685.) - Tom Copeland, Oct 21 2011

More terms from Eric Rains, Apr 02 2005

A074059 Dimension of the cohomology ring of the moduli space of n-pointed curves of genus 0 satisfying the associativity equations of physics (also known as the WDVV equations).

Original entry on oeis.org

1, 1, 2, 7, 34, 213, 1630, 14747, 153946, 1821473, 24087590, 352080111, 5636451794, 98081813581, 1843315388078, 37209072076483, 802906142007946, 18443166021077145, 449326835001457846, 11572432709175470807, 314160322966817351938, 8965995574654847062469

Offset: 1

Margaret A. Readdy, Aug 16 2002

nonn

			From _Paul D. Hanna_, Sep 24 2010: (Start)
E.g.f.: x + x^2/2! + 2*x^3/3! + 7*x^4/4! + 34*x^5/5! + 213*x^6/6! +...
The series reversion of the e.g.f. begins:
x - x^2/2 + x^3/6 - x^4/12 + x^5/20 - x^6/30 + x^7/42 - x^8/56 +... (End)

Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
V. M. Buchstaber and A. P. Veselov, Differential algebra of polytopes and inversion formulas, arXiv:2402.07168 [math.CO], 2024. See p. 9.
Brian Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), #07.3.7.
I. P. Goulden, S. Litsyn, and V. Shevelev, On a Sequence Arising in Algebraic Geometry, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.7.
S. Keel, Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545-574.
M. Kontsevich and Y. Manin, Quantum cohomology of a product, (with Appendix by R. Kaufmann), Inv. Math. 124, f. 1-3 (1996) 313-339.
Curtis T. McMullen, Moduli spaces in genus zero and inversion of power series, (2012).
Margaret Readdy, The pre-WDVV ring of physics and its topology, arXiv:math/0511420 [math.CO], The Ramanujan Journal, Special issue on the Number Theory and Combinatorics in Physics, 10 (2005), 269-281.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.

Row sums of triangle A074060.

series(exp(LambertW(-exp(-2)*(2+x))+2)-1,x,30): A:=simplify(%,symbolic): A074059:=n->n!*coeff(A,x,n): # Gessel

max = 19; $Assumptions = x > 0; (Series[ Exp[2 + ProductLog[-1, -(x+2)/E^2]] - 1, {x, 0, 19}] // CoefficientList[#, x] &) * Range[0, 19]! // Rest (* Jean-François Alcover, Jun 20 2013 *)

{a(n)=if(n<1,0,n!*polcoeff(serreverse(x-sum(k=2,n,(-x)^k/(k*(k-1)))+x*O(x^n)),n))} \\ Paul D. Hanna, Sep 24 2010

The exponential generating function A = A(x) = sum_{n>=1} a(n) x^n/n! satisfies the equation (1+A)log(1+A) = 2A-x. Explicitly, 1+A(x) = exp(2+W(e^(-2)(2+x))), where W is Lambert's W-function. - Ira M. Gessel, Dec 15 2005
E.g.f.: Series_Reversion[ x - Sum_{n>=2} (-x)^n/(n(n-1)) ]. - Paul D. Hanna, Sep 24 2010
Let h(x) = 1/(1-log(1+x)), then a(n) = ((h(x)*d/dx)^n)x evaluated at x=0, i.e., A(x) = exp(x*a(.)) = exp(x*h(u)*d/du) u, evaluated at u=0. Also, dA(x)/dx = h(A(x)). - Tom Copeland, Sep 06 2011
An o.g.f. is provided by the integral from w=0 to infinity of exp(-2w) * (1+z*w)^((1+z*w)/z). - Tom Copeland, Sep 09 2011
E.g.f. = -1/{1+W[-(2+x) exp(-2)]} with W(x) the Monir branch of the Lambert W fct. defined in A135338 and offset 0. - Tom Copeland, Oct 05 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x)*exp(-x)*d/dx. Cf. A061356. - Peter Bala, Dec 08 2011
a(n) ~ n^(n-1) / (exp(1)*(exp(1)-2))^(n-1/2). - Vaclav Kotesovec, Oct 05 2013
a(1) = 1; a(n) = a(n-1) + Sum_{k=2..n-1} binomial(n-1,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Aug 28 2020

More terms from Ira M. Gessel, Dec 15 2005

a(20)-a(22) from Stefano Spezia, Feb 14 2024

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User: Margaret A. Readdy

A256161 Triangle of allowable Stirling numbers of the second kind a(n,k).

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A074060 Graded dimension of the cohomology ring of the moduli space of n-pointed stable curves of genus 0 satisfying the associativity equations of physics (also known as the WDVV equations).

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A074059 Dimension of the cohomology ring of the moduli space of n-pointed curves of genus 0 satisfying the associativity equations of physics (also known as the WDVV equations).

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