A268172 -id:A268172 - cp's OEIS frontend

A292085 Number A(n,k) of (unlabeled) rooted trees with n leaf nodes and without unary nodes or outdegrees larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 4, 3, 0, 1, 1, 2, 5, 9, 6, 0, 1, 1, 2, 5, 11, 23, 11, 0, 1, 1, 2, 5, 12, 30, 58, 23, 0, 1, 1, 2, 5, 12, 32, 80, 156, 46, 0, 1, 1, 2, 5, 12, 33, 87, 228, 426, 98, 0, 1, 1, 2, 5, 12, 33, 89, 251, 656, 1194, 207, 0

Offset: 1

Alois P. Heinz, Sep 08 2017

			:               T(4,3) = 4             :
:                                      :
:       o       o         o       o    :
:      / \     / \       / \     /|\   :
:     o   N   o   o     o   N   o N N  :
:    / \     ( ) ( )   /|\     ( )     :
:   o   N    N N N N  N N N    N N     :
:  ( )                                 :
:  N N                                 :
:                                      :
Square array A(n,k) begins:
  1,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,   2,   2,   2,   2,   2,   2, ...
  0,  2,   4,   5,   5,   5,   5,   5, ...
  0,  3,   9,  11,  12,  12,  12,  12, ...
  0,  6,  23,  30,  32,  33,  33,  33, ...
  0, 11,  58,  80,  87,  89,  90,  90, ...
  0, 23, 156, 228, 251, 258, 260, 261, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index entries for sequences related to rooted trees

Columns k=1-10 give: A063524, A001190, A268172, A292210, A292211, A292212, A292213, A292214, A292215, A292216.
Main diagonal gives A000669.
Cf. A244372, A288942, A292086.

b:= proc(n, i, v, k) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n

b[n_, i_, v_, k_] := b[n, i, v, k] = If[n == 0, If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0, If[v == n, 1, Sum[Binomial[A[i, k] + j - 1, j]*b[n - i*j, i - 1, v - j, k], {j, 0, Min[n/i, v]}]]]];
A[n_, k_] := A[n, k] = If[n < 2, n, Sum[b[n, n + 1 - j, j, k], {j, 2, Min[n, k]}]];
Table[Table[A[n, 1 + d - n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

A(n,k) = Sum_{j=1..k} A292086(n,j).

A268163 Number of labeled binary-ternary rooted non-planar trees, indexed by number of leaves.

Original entry on oeis.org

0, 1, 1, 4, 25, 220, 2485, 34300, 559405, 10525900, 224449225, 5348843500, 140880765025, 4063875715900, 127418482316125, 4314607214417500, 156920190449147125, 6100643259005795500, 252476539015516440625, 11081983532721088487500, 514215436341672155715625

Offset: 0

Murray R. Bremner, Jan 27 2016

nonn

This can also be interpreted as the number of multilinear monomials of degree n in a nonassociative algebra with an (anti)commutative binary operation and a completely (skew-)symmetric ternary operation; the number of variables in the monomial corresponds to the number of leaves in the tree.

This sequence also enumerates a certain class of Feynman diagrams; see the references, links, and crossrefs below.

			For n = 4 and using the monomial interpretation, the 25 multilinear monomials of degree 4 are as follows, where [-,-] is the binary operation and (-,-,-) is the ternary operation:
[[[a,b],c],d], [[[a,b],d],c], [[[a,c],b],d], [[[a,c],d],b], [[[a,d],b],c], [[[a,d],c],b], [[[b,c],a],d], [[[b,c],d],a], [[[b,d],a],c], [[[b,d],c],a], [[[c,d],a],b], [[[c,d],b],a], [[a,b],[c,d]], [[a,c],[b,d]], [[a,d],[b,c]], [(a,b,c),d], [(a,b,d),c], [(a,c,d),b], [(b,c,d),a], ([a,b],c,d), ([a,c],b,d), ([a,d],b,c), ([b,c],a,d), ([b,d],a,c), ([c,d],a,b).

J. Bedford, On Perturbative Field Theory and Twistor String Theory, Ph.D. Thesis, 2007, Queen Mary, University of London.
B. Feng and M. Luo, An introduction to on-shell recursion relations, Review Article, Frontiers of Physics, October 2012, Volume 7, Issue 5, pp. 533-575.
K. Kampf, A new look at the nonlinear sigma model, 17th International Conference in Quantum Chromodynamics (QCD 14), Nuclear and Particle Physics Proceedings, Volumes 258-259, January-February 2015, pp. 86-89.
M. L. Mangano and S. J. Parke, Multi-parton amplitudes in gauge theories, Physics Reports, Volume 200, Issue 6, February 1991, pp. 301-367.

Alois P. Heinz, Table of n, a(n) for n = 0..400.
J. Bedford, On Perturbative Field Theory and Twistor String Theory, arXiv:0709.3478 [hep-th], 2007, (see page 23).
D. Carmi, TeV scale strings and scattering amplitudes at the LHC, arXiv:1109.5161 [hep-th], 2011-2015, (see page 23).
B. Feng, M. Luo, An introduction to on-shell recursion relations, arXiv:1111.5759 [hep-th], 2011-2012, (see page 2).
K. Kampf, A new look at the nonlinear sigma model (see pages 7 and 13).
M. L. Mangano, S. J. Parke, Multi-parton amplitudes in gauge theories, arXiv:hep-th/0509223, 2005, (see page 5).
G. Travaglini, Harmony of (super) form factors (see page 3).
A. Volovich, Yang-Mills amplitudes and twistor string theory (see bottom of page 1).
James Christopher Whitehead, The Production of Pairs of Isolated Photons at Higher Orders in QCD, Ph. D. Thesis, Durham University (UK, 2021).
Wikipedia, Feynman diagram

Cf. A001147. The number of labeled binary rooted non-planar trees.
Cf. A025035. The number of labeled ternary rooted non-planar trees.
Cf. A268172. The corresponding number of unlabelled trees.
Cf. A005411. Number of non-vanishing Feynman diagrams of order 2n for the electron or the photon propagators in quantum electrodynamics.
Cf. A005412. Number of non-vanishing Feynman diagrams of order 2n for the vacuum polarization (the proper two-point function of the photon) and for the self-energy (the proper two-point function of the electron) in quantum electrodynamics (QED).
Cf. A005413. Number of non-vanishing Feynman diagrams of order 2n+1 for the electron-electron-photon proper vertex function in quantum electrodynamics (QED).
Cf. A005414. Feynman diagrams of order 2n with vertex skeletons.
Other sequences related to Feynman diagrams: A115974, A122023, A167872, A214298, A214299.
Cf. A000311.

with(combinat):
b:= proc(n, i, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or nAlois P. Heinz, Jan 28 2016
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [0, 1$2][n+1],
       ((24*n-36)*a(n-1)+(3*n-5)*(3*n-7)*a(n-2))/11)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 28 2016

a[0]=0; a[1]=1; a[2]=1; a[n_]:=a[n]=(12(2n-3)a[n-1]+(3n-5)(3n-7)a[n-2])/11; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)

a(n) = ((24*n-36)*a(n-1)+(3*n-5)*(3*n-7)*a(n-2))/11 for n>2. - Alois P. Heinz, Jan 28 2016
Because of Koszul duality for operads, the exponential generating function is the compositional inverse of the power series x-x^2/2-x^3/6 (email of Vladimir Dotsenko to Murray R. Bremner, Jan 28 2016).
a(n) ~ sqrt(9-4*sqrt(3)) * ((12+9*sqrt(3))/11)^n * n^(n-1) / (3 * exp(n)). - Vaclav Kotesovec, Feb 24 2016

A276277 Association types for monomials with n arguments in an algebra with two binary operations, one commutative, one noncommutative.

Original entry on oeis.org

1, 2, 6, 25, 111, 540, 2736, 14396, 77649, 427608, 2392866, 13570386, 77815161, 450418536, 2628225684, 15443406868, 91301938365, 542704450806, 3241411991712, 19443499011192, 117084197728737, 707532791560272, 4289252607915012, 26078561954153631

Offset: 1

Murray R. Bremner, Aug 26 2016

nonn

a(n) is the number of complete rooted binary trees with n leaves in which the internal nodes are labeled either white or black; the two children (subtrees) of a white node have no specified orientation, but the two children (subtrees) of a black node are labeled left and right. Thus the notion of isomorphism for these trees is partly planar (for the black nodes) and partly abstract (for the white nodes).

Finding a recurrence relation is an easy exercise. Finding an exact formula is probably very difficult or even impossible: compare the OEIS page for A001190 (Wedderburn-Etherington numbers).

			For n = 4 the 25 association types are as follows, where * is commutative and # is noncommutative; some assumptions have been made regarding the order of the factors for the commutative operation:
( ( X * X ) * X ) * X,
( ( X # X ) * X ) * X,
( ( X * X ) # X ) * X,
( ( X # X ) # X ) * X,
( X # ( X * X ) ) * X,
( X # ( X # X ) ) * X,
( X * X ) * ( X * X ),
( X * X ) * ( X # X ),
( X # X ) * ( X # X ),
( ( X * X ) * X ) # X,
( ( X # X ) * X ) # X,
( ( X * X ) # X ) # X,
( ( X # X ) # X ) # X,
( X # ( X * X ) ) # X,
( X # ( X # X ) ) # X,
( X * X ) # ( X * X ),
( X * X ) # ( X # X ),
( X # X ) # ( X * X ),
( X # X ) # ( X # X ),
X # ( ( X * X ) * X ),
X # ( ( X # X ) * X ),
X # ( ( X * X ) # X ),
X # ( ( X # X ) # X ),
X # ( X # ( X * X ) ),
X # ( X # ( X # X ) ).

F. Bagherzadeh, M. R. Bremner, S. Madariaga, Jordan trialgebras and post-Jordan algebras, arXiv:1611.01214 [math.RA], 2016.
Murray Bremner, Martin Markl, Distributive laws between the Three Graces, arXiv:1809.08191 [math.AT], 2018.

Cf. A001190, A000108, A226909, A268172.

BWT := table():
BWT[ 1 ] := 1:
for arity from 2 to 24 do
  BWT[ arity ] := 0:
  # commutative operation
  for i to floor((arity-1)/2) do
    BWT[ arity ] := BWT[ arity ] + ( BWT[arity-i] * BWT[i] )
  od:
  if arity mod 2 = 0 then
    BWT[ arity ] := BWT[ arity ] + binomial( BWT[arity/2]+1, 2 )
  fi:
  # noncommutative operation
  for i to arity-1 do
    BWT[ arity ] := BWT[ arity ] + ( BWT[arity-i] * BWT[i] )
  od
od:
seq(BWT[ n ], n=1..24);

BWT[1] = 1; For[arity = 2, arity <= 24, arity++, BWT[arity] = 0; (* commutative operation *) For[i = 1, i <= Floor[(arity-1)/2], i++, BWT[arity] = BWT[arity] + (BWT[arity-i]*BWT[i])]; If[EvenQ[arity], BWT[arity] = BWT[arity] + Binomial[BWT[ arity/2]+1, 2]]; (* non commutative operation *) For[i = 1, i <= arity-1, i++, BWT[arity] = BWT[arity] + (BWT[arity-i]*BWT[i])]];
Table[BWT[n], {n, 1, 24}] (* Jean-François Alcover, Feb 15 2019, from Maple *)

cp's OEIS Frontend

A292085 Number A(n,k) of (unlabeled) rooted trees with n leaf nodes and without unary nodes or outdegrees larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A268163 Number of labeled binary-ternary rooted non-planar trees, indexed by number of leaves.

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A276277 Association types for monomials with n arguments in an algebra with two binary operations, one commutative, one noncommutative.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Maple

Mathematica