cp's OEIS Frontend

A139002 Weights of Connes-Moscovici Hopf subalgebra.

%I A139002 #113 Jun 16 2025 08:51:11
%S A139002 1,1,1,1,1,1,3,1,1,3,1,1,4,4,3,6,1,1,6,3,4,4,1,1,3,1,5,15,5,5,10,10,
%T A139002 10,10,15,10,1,1,10,15,10,10,10,10,5,5,15,5,1,3,1,1,4,4,3,6,1,6,36,18,
%U A139002 24,24,6,6,18,6,15,15,45,45,15,15,15,15,10,20,60,20,10,60,20,15,45,15,1,1
%N A139002 Weights of Connes-Moscovici Hopf subalgebra.
%C A139002 Gives multiplicity for tree shapes in "naturally grown" forests of rooted trees (from file for CM(t) referred to in Broadhurst).
%C A139002 A refinement of the enumeration of the trees of the first few forests in terms of planar rather than nonplanar rooted trees is presented on p. 21 of the Munthe-Kaas and Lundervold paper. - _Tom Copeland_, Jul 16 2018 (The refinement is presented also in Lundervold and on p. 35 of the Lundervold and Munthe-Kaas paper. - _Tom Copeland_, Jul 21 2021)
%C A139002 Enumerates the elementary differentials of the Butcher group that Cayley showed are in bijection with these nonplanar rooted trees when considering multivariable vector functions. When considering a scalar function of one independent variable, the associated differentials are no longer in bijection with the planar trees and are enumerated by A139605. Two nonplanar trees are considered equivalent if the branches of one may be rotated about its nodes to match those of the other. - _Tom Copeland_, Jul 21 2021
%D A139002 J. Butcher, Numerical Methods for Ordinary Differential Equations, 3rd Ed., Wiley, 2016, Table 310(II) on p. 165.
%D A139002 E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration - Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd Ed., Springer, 2006, pp. 52 and 53.
%H A139002 S. Agarwala and C. Delaney, <a href="http://arxiv.org/abs/1302.4004">Generalizing the Connes-Moscovici Hopf algebra to include all rooted trees</a>, arXiv:1302.4004 [math-ph], 2015, pp. 2, 9, and 21.
%H A139002 C. Bergbauer and D. Kreimer, <a href="https://arxiv.org/abs/hep-th/0506190">Hopf algebras in renormalization theory: Locality and Dyson-Schwinger equations from Hochschild cohomology</a>, arXiv:hep-th/0506190 [hep-th], 2005-2006, p. 14.
%H A139002 D. J. Broadhurst and D. Kreimer, <a href="http://arXiv.org/abs/hep-th/9810087">Renormalization automated by Hopf algebra</a>, arXiv:hep-th/9810087 [hep-th], 1998.
%H A139002 C. Brouder, <a href="https://citeseerx.ist.psu.edu/pdf/a51b88800479e028b0f174aadbeb81dd643990be">Trees, renormalization, and differential equations</a>, BIT Numerical Mathematics, 44: 425-438, 2004, p. 434.
%H A139002 F. Chapoton, <a href="http://arxiv.org/abs/math/0209104">Rooted trees and an exponential-like series</a>, arXiv:math/0209104 [math.QA], 2002.
%H A139002 A. Connes and D. Kreimer, <a href="https://arxiv.org/abs/hep-th/9808042">Hopf algebras, renormalization, and noncommutative geometry</a>, arXiv:hep-th/9808042 [hep-th], 1998, pp. 21 and 28.
%H A139002 Tom Copeland, <a href="http://tcjpn.spaces.live.com/default.aspx">A Walk in the Woods with Cayley and Comtet</a>, 2008.
%H A139002 Tom Copeland, <a href="http://tcjpn.wordpress.com/2008/06/12/mathemagical-forests/">Mathemagical Forests</a>, 2008.
%H A139002 W. Dugan, L. Foissy, and K. Yeats, <a href="https://arxiv.org/abs/2209.06576">Sequences of Trees and Higher-Order Renormalization Group Equations</a>, arXiv:2209.06576 [math.CO], 2023, p. 4.
%H A139002 L. Foissy, <a href="http://arXiv.org/abs/0707.1204">Faa di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations</a>, arXiv:0707.1204 [math.RA], 2007, p. 2.
%H A139002 E. Getzler, <a href="https://arxiv.org/abs/math/0404003">Lie theory for nilpotent L-infinity algebras</a>, arXiv:math/0404003 [math.AT], 2004-2007, p. 22.
%H A139002 M. Ginocchio, <a href="https://doi.org/10.1007/BF00750066">Universal expansion of the powers of a derivation</a>, Lett. Math. Phys. 34, 343-364, 1995, see table on p. 364.
%H A139002 T. Krajewski and T. Martinetti, <a href="https://arxiv.org/abs/0806.4309">Wilsonian renormalization, differential equations and Hopf algebras</a>, arXiv:0806.4309 [hep-th], 2008, p. 14.
%H A139002 D. Kreimer, <a href="https://www2.mathematik.hu-berlin.de/~kreimer/wp-content/uploads/SkriptRGE.pdf">Renormalization and Renormalization Group</a>, lecture notes by L. Klaczynski of class lectures by Dirk Kreimer, 2012, p. 16.
%H A139002 A. Lundervold, <a href="https://alexander.lundervold.com/assets/files/phdthesis.pdf">Lie-Butcher series and geometric numerical integration on manifolds</a>, Ph.D. thesis, Dept. of Math., Univ. of Bergen, 2011, pp. 8 and 10.
%H A139002 A. Lundervold and H. Munthe-Kaas, <a href="https://arxiv.org/abs/1112.4465">On algebraic structures of numerical integration on vector spaces and manifolds</a>, arXiv:1112.4465 [math.QA], 2013. p. 6.
%H A139002 Mathoverflow, <a href="https://mathoverflow.net/questions/41039/formula-for-n-th-iteration-of-dx-dt-bx">Formula for n-th iteration of dx/dt=B(x)</a>, a question on MathOverflow posed by the user resolvent and answered by Tom Copeland, 2021.
%H A139002 H. Munthe-Kaas and A. Lundervold, <a href="https://arxiv.org/abs/1203.4738">On post-Lie algebras, Lie-Butcher series and moving frames</a>, arXiv:1203.4738 [math.NA], 2013.
%F A139002 The table on p. 364 of Ginocchio contains the Connes-Moscovici weights correlated with associated derivatives D^k. From the relation of this entry to A139605 and A145271, the action of the weighted differentials on an exponential is associated with the operation exp(x g(u)D_u) e^(ut) = e^(t H^{(-1)}(H(u)+x)) with g(x) = 1/D(H(x)) and H^{(-1)} the compositional inverse of H. With H^{(-1)}(x) = -log(1-x), the inverse about x=0 is H(x) = 1-e^(-x), giving g(x) = e^x and the resulting action e^(-t log(1-x)) = (1-x)^(-t) for u=0, an e.g.f. for the unsigned Stirling numbers of the first kind A008275 and A048994. Consequently, summing the Connes-Moscovici weights over each associated derivative gives these Stirling numbers. E.g., the fifth row in the examples reduces to (1+3+1+1) D + (4+4+3) D^2 + 6 D^3 + D^4 = 6 D + 11 D^2 + 6 D^3 + D^4. - _Tom Copeland_, Jul 14 2021
%e A139002 From _Tom Copeland_, Dec 06 2017: (Start)
%e A139002 The number of distinct rooted tree types, or shapes, with n nodes is given by A000081(n+1), so the multiplicities for the tree shapes of the forests of naturally grown trees given here may be grouped according to A000081. For example, A000081(5)=4 corresponds to the four tree types depicted in Fig. 6 of Mathemagical Forests with four nodes, or vertices, with the four multiplicities (1,1,3,1); A000081(4)=2 corresponds to the two tree types depicted in Fig. 3 with three nodes and the two multiplicities (1,1); A000081(3)=1, with one tree type with two nodes and multiplicity (1); and A000081(2)=1, with one tree type with one node and multiplicity (1). Then the sequence here begins (1)(1)(1,1)(1,1,3,1).
%e A139002 First few rows (with last row reordered according to Fig. 7 of Mathemagical Forests):
%e A139002   1
%e A139002   1
%e A139002   1, 1
%e A139002   1, 1, 3, 1
%e A139002   1, 1, 3, 4, 4, 3, 6, 1, 1
%e A139002 This last row corresponds to the one listed in Broadhurst as
%e A139002   1, 3, 1, 1, 4, 4, 3, 6, 1.
%e A139002 (End)
%t A139002 nmax = 7;
%t A139002 SetAttributes[t, Orderless];
%t A139002 size[tree_] := Count[tree, _, All];
%t A139002 lst = {{t[]}};
%t A139002 forests[0, 0] = {{}}; forests[_?Positive, 0] = {}; forests[_?Negative, _] = {};
%t A139002 forests[n_, k_] := forests[n, k] = With[{tree = Flatten[lst][[k]]}, Join[Append[tree] /@ forests[n - size@tree, k], forests[n, k-1]]];
%t A139002 Do[AppendTo[lst, t @@@ forests[n-1, Length[Flatten@lst]]], {n, 2, nmax}];
%t A139002 assoc = Association[{# -> 0} & /@ Flatten@lst];
%t A139002 assoc[t[]] = 1;
%t A139002 Do[assoc[Insert[tree, t[], Append[Most@p, 1]]] += assoc[tree], {n, 2, nmax}, {tree, lst[[n-1]]}, {p, Position[tree, t]}];
%t A139002 Last /@ Normal@assoc (* _Andrey Zabolotskiy_, Mar 15 2024 *)
%Y A139002 Cf. A000081, A008275, A048994, A139605, A145271, A206495.
%Y A139002 A206496 is a permutation of this sequence.
%K A139002 nonn,tabf
%O A139002 1,7
%A A139002 _Tom Copeland_, May 31 2008