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A133314 Coefficients of list partition transform: reciprocal of an exponential generating function (e.g.f.). 74
1, -1, -1, 2, -1, 6, -6, -1, 8, 6, -36, 24, -1, 10, 20, -60, -90, 240, -120, -1, 12, 30, -90, 20, -360, 480, -90, 1080, -1800, 720, -1, 14, 42, -126, 70, -630, 840, -420, -630, 5040, -4200, 2520, -12600, 15120, -5040, -1, 16, 56, -168, 112, -1008, 1344, 70 (list; graph; refs; listen; history; text; internal format)



The list partition transform of a sequence a(n) for which a(0)=1 is illustrated by:

b_0 = 1

b_1 = -a_1

b_2 = -a_2 + 2 a_1^2

b_3 = -a_3 + 6 a_2 a_1 - 6 a_1^3

b_4 = -a_4 + 8 a_3 a_1 + 6 a_2^2 - 36 a_2 a_1^2 + 24 a_1^4

... .

The unsigned coefficients are A049019 with a leading 1. The sign is dependent on the partition as evident from inspection (replace a_n's by -1).

Expressed umbrally, i.e., with the umbral operation (a.)^n := a_n,

  exp(a.x) exp(b.x) = exp[(a.+b.)x] = 1; i.e., (a.+b.)^n = 1 for n=0 and 0 for all other values of n.

Expressed recursively,

b_0 = 1, b_n = -Sum_{j=1..n} binomial(n,j) a_j b_{n-j}; which is conditionally self-inverse, i.e., the roles of a_k and b_k may be reversed with a_0 = b_0 = 1.

Expressed in matrix form, b_n form the first column of B = matrix inverse of A .

A = Pascal matrix diagonally multiplied by a_n, i.e., A_{n,k} = binomial(n,k)* a_{n-k}.

Some examples of reciprocal pairs of sequences under these operations are:

1) A084358 and -A000262 with the first term set to 1.

2) (1,-1,0,0,...) and (0!,1!,2!,3!,...) with the unsigned associated matrices A128229 and A094587.

3) (1,-1,-1,-1,...) and A000670.

4) A000110 and A000587.

5) (1,-2,-2,0,0,0,...) and (0! c_1,1! c_2,2! c_3,3! c_4,...) where c_n = A000129(n) with the associated matrices A110327 and A110330.

6) (1,-2,2,0,0,0,...) and (1!,2!,3!,4!,...).

7) Sequences of rising and signed lowering factorials form reciprocal pairs where a_n = (-1)^n m!/(m-n)! and b_n = (m-1+n)!/(m-1)! for m=0,1,2,... .

Denote the action of the list partition transform on the sequence a. or an invertible matrix M by LPT(a.) = b. or LPT(M)= M^(-1).

If the matrix equation M = exp(T) also holds, then exp[a.*T]*exp[b.*T] = exp[(a.+b.)*T] = I, the identity matrix, because (a.+b.)^n = delta_n, the Kronecker delta with delta_n = 1 and delta_n = 0 otherwise, i.e., (0)^n = delta_n.

Therefore, [exp(a.*T)]^(-1) = exp[b.*T] = exp[LPT(a.)*T] = LPT[exp(a.*T)].

The fundamental Pascal (A007318), unsigned Lah (A105278) and associated Laguerre matrices can be generated by exponentiation of special infinitesimal matrices (see A132440, A132710 and A132681) such that finding LPT(a.) amounts to multiplying the k'th diagonal of the fundamental matrices by a_k for every diagonal followed by matrix inversion and then extraction of the b_n factors from the first column (simplest for the Pascal formulas above).

Conversely, the inverses of matrices formed by diagonally multiplying the three fundamental matrices by a_k are given by diagonally multiplying the fundamental matrices by b_k.

If LPT(M) is defined differently as application of the top formula to a_n = M^n, then b_n = (-M)^n and the formalism could even be applied to more general sequences of matrices M., providing the reciprocal of exp[t*M.].

The group of fundamental lower triangular matrices M = exp(T) such that LPT[exp(a.*T)] = exp[LPT(a.)*T] = [exp[a.*T]]^(-1) are obtained by infinitesimal generator matrices of the form T =


  t(0), 0;

  0, t(1), 0;

  0, 0, t(2), 0;

  0, 0, 0, t(3), 0;

  ... .

T^m has trivially vanishing terms except along the m'th subdiagonal, which is a sequence of generalized factorials:

[ t(0)*t(1)...t(m-2)*t(m-1), t(1)*t(2)...t(m-1)*t(m), t(2)*t(3)...t(m)*t(m+1), ... ].

Therefore the principal submatrices of T (given by setting t(j) = 0 for j > n-1) are nilpotent with at least [Tsub_n]^(n+1) = 0.

The general group of matrices GM[a.] = exp[a.*T] can also be obtained through diagonal multiplication of M = exp(T) by the sequence a_n, as in the Pascal matrix example above and their inverses by diagonal multiplication by b. = LPT(a.).

Weighted-mappings interpretation for the top partition equation:

Given n pre-nodes (Pre) and k post-nodes (Post), each Pre is connected to only one Post and each Post has at least one Pre connected to it (surjections or onto functions/maps). Weight each Post by -a_m where m is the number of connections to the Post.

Weight each map by the product of the Post weights and multiply by the number of maps that share the same connectivity. Sum over the possible mappings for n Pre. The result is b_n.

E.g., b_3 = [ 3 Pre to 1 Post ] + [ 3 Pre to 2 Post ] + [ 3 Pre to 3 Post ]

= [1 map with 1 Post with 3 connections] + [ 6 maps with 1 Post with 2 connections and 1 Post with 1 connection] + [6 maps with 3 Post with 1 connection each]

= -a_3 + 6 * [-a_2*(-a_1)] + 6 * [-a_1*(-a_1)*(-a_1)].

See A263633 for the complementary formulation for the reciprocal of o.g.f.s rather than e.g.f.s and computations of these partition polynomials as Gram determinants. - Tom Copeland, Dec 04 2016

The coefficients of the partition polynomials enumerate the faces of the convex, bounded polytopes called the permutahedra, or permutohedra, and the absolute value of the sum of the coefficients gives the Euler characteristic of unity for each polytope; i.e., the absolute value of the sum of each row of the array is unity. In addition, the signs of the faces alternate with dimension, and the coefficients of faces with the same dimension for each polytope have the same sign. - Tom Copeland, Nov 13 2019

With the fundamental matrix chosen to be the lower triangular Pascal matrix M, the matrix MA whose n-th diagonals are multiplied by a_n (i.e., MA_{i,j} = PM_{i,j} * a_{i-j}) gives a matrix representation of the e.g.f. associated to the Appell polynomial sequence defined by e^{a.t}e^{xt}= e^{(a.+x)t} = e^{A.(x)t} where umbrally (A.(x))^n = A_n(x) = (a. + x)^n = sum_{k=0..n} binomial(n,k) a_k x^{n-k} are the associated Appell polynomials. Left multiplication of the column vector (1,x,x^2,..) by MA gives the Appell polynomial sequence, and multiplication of the two e.g.f.s e^{a.t} and e^{b.t} corresponds to multiplication of their respective matrix representations MA and MB. Forming the reciprocal of an e.g.f. corresponds to taking the matrix inverse of its matrix representation as noted above. A263634 gives an associated modified Pascal matrix representation of the raising operator for the Appell sequence. - Tom Copeland, Nov 13 2019

The diagonal of MA consists of all ones. Let MAN be the truncated square submatrix of MA containing the coefficients of the first N Appell polynomials A_k=(a.+x)^k = Sum(j=0 to k) MAN(k,j) x^j. Then by the Cayley-Hamilton theorem (I-MAN)^N = 0; therefore, MAN^(-1) = Sum(k=1 to N) binomial(N,k) (-MAN)^{k-1} = MBN, the inverse of MAN, containing the coefficients of the first N rows of the Appell polynomials B_k(x) = (b. + x)^k = Sum(j=0 to k) MBN(k,j) x^j, which are the umbral compositional inverses of the Appell row polynomials A_k(x) of MAN; that is, A_k(B.(x)) = x^k = B_k(A.(x)), where, e.g., (A.(x))^k = A_k(x). - Tom Copeland, May 13 2020

The use of the term 'list partition transform' resulted from one of my first uses of these partition polynomials in relating A000262 to A084358 with their simple e.g.f.s. Other appropriate names would be the permutahedra polynomials since they are refined Euler characteristics of the permutahedra or the reciprocal polynomials since they give the multiplicative inverses of e.g.f.s with a constant of 1. - Tom Copeland, Oct 09 2022


Vincenzo Librandi, Table of n, a(n) for n = 0..250

M. Aguiar, Math 7410: Lie Combinatorics and Hyperplane Arrangements, notes made by D. Mehrle for course taught by Aguiar at Cornell, 2016, p. 49.

M. Aguiar and F. Ardila, The algebraic and combinatorial structure of generalized permutahedra, MSRI Summer School July 19, 2017.

M. Aguiar and F. Ardila, Hopf monoids and generalized permutahedra, arXiv:1709.07504 [math.CO], p. 5, 2017.

N. Arkani-Hamed, Y. Bai, S. He, and G. Yan, Scattering forms and the positive geometry of kinematics, color, and the worldsheet , arXiv:1711.09102 [hep-th], 2017.

J. Bergström and S. Minabe, On the cohomology of the Losev-Manin moduli space, arXiv:1108.0338 [math.AG], (cf. p. 1), 2013.

Z. Bern, J.Carrasco, M. Chiodaroli, H. Johansson, and R. Roiban , The Duality Between Color and Kinematics and its Applications, arXiv preprint arXiv:1909.01358 [hep-th], 2019.

L. Berry, S. Forcey, M. Ronco, and P. Showers, Polytopes and Hopf algebras of painted trees: Fan graphs and Stellohedra, arXiv:1608.08546 [math.CO], 2018.

Tom Copeland, Bijective mapping between face polytopes of permutohedra and partitions of integers, Math StackExchange question, 2016.

Tom Copeland, Compilation of OEIS Partition Polynomials A133314, A134685, A145271, A356144, and A356145, 2022.

Tom Copeland, In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms, 2019.

Karl-Dieter Crisman, The Borda Count, the Kemeny Rule, and the permutahedron, preprint, 2014.

Karl-Dieter Crisman, The Borda Count, the Kemeny Rule, and the permutahedron, in: Karl-Dieter Crisman and Michael A. Jones (eds.), The Mathematics of Decisions, Elections, and Games, Contemporary Mathematics, AMS, Vol. 624, 2014, pp. 101-134.

R. Da Rosa, D. Jensen, and D. Ranganathan, Toric graph associahedra and compactifications of M_(0,n), arXiv:1411.0537 [math.AG], 2015.

Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.

S. Forcey, The Hedra Zoo

S. Franco and A. Hasan, Graded Quivers, Generalized Dimer Models and Toric Geometry , arXiv preprint arXiv:1904.07954 [hep-th], 2019.

M. Futaki and K. Ueda, Tropical coamoeba and torus-equivariant homological mirror symmetry for the projective space , arXiv preprint arXiv:1001.4858 [math.SG], 2010-2014.

X. Gao, S. He, Y. Zhan, Labelled tree graphs, Feynman diagrams and disk integrals , arXiv:1708.08701 [hep-th], 2017.

A. Hasan, Physics and Mathematics of Graded Quivers, dissertation, Graduate Center, City University of New York, 2019.

D. Karp, D. Ranganathan, P. Riggins, and U. Whitcher, Toric Symmetry of CP^3, arXiv:1109.5157 [math.AG], 2011.

R. Kaufmann and Y. Zhang, Permutohedral structures on E2-operads, arXiv preprint arXiv:1602.08247 [math.AT], 2016.

M. Lin, Graph Cohomology, 2016.

J. Loday, The Multiple Facets of the Associahedron

MathOverflow, Analogue of conic sections for the permutohedra, associahedra, and noncrossing partitions, an MO question posed by T. Copeland, 2017.

W. Norledge and A.  Ocneanu, Hopf monoids, permutohedral cones, and generalized retarded functions, arXiv preprint arXiv:1911.11736 [math.CO], 2020.

P. Olver, The canonical contact form, p. 9.

V. Pilaud, The Associahedron and its Friends, presentation for Séminaire Lotharingien de Combinatoire, April 4 - 6, 2016.

J . Pitman and R. Stanley, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, arxiv: 9908029 [math.CO], 1999.

A. Postnikov, Positive Grassmannian and Polyhedral Subdivisions, arXiv:1806.05307 [math.CO], (cf. p. 17), 2018.

D. Ranganathan, Gromov-Witten Theory of Blowups of Toric Threefolds, senior thesis, Harvey Mudd College, 2012.

D. Ranganathan, Gromov-Witten Theory of Blowups of Toric Threefolds (poster), poster for senior thesis, Harvey Mudd College, 2012.

E. Schröder, Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Mathematische Annalen vol. 2, 317-365, 1870, (bottom of p. 342, see Stewart below for a translation).

G. Stewart, On infinitely many algorithms for solving equations, 1993, (translation into English of the Schröder paper above, see top of p. 31 for differently normalized partition polynomials of this entry).


b_{n-1} = (1/n)(d/da(1))p_n[a_1, a_2, ..., a_n] where p_n are the row partition polynomials of the cumulant generator A127671. - Tom Copeland, Oct 13 2012

(E.g.f. of matrix B) = (e.g.f. of b)·exp(xt) = exp(b.t)·exp(xt) = exp(xt)/exp(a.t) = (e.g.f. of A^(-1)) and (e.g.f. of matrix A) = exp(a.t)·exp(xt) = exp(xt)/exp(b.t) = (e.g.f. of B^(-1)), where the umbral evaluation of exp(b.t) = Sum{n >= 0} (b.t)^n / n! = Sum_{n >= 0} b_n t^n / n! is understood in the denominator. These e.g.f.s define Appell sequences of polynomials. - Tom Copeland, Mar 22 2014

Sum of the n-th row is (-1)^n. - Peter Luschny, Sep 18 2015

The unsigned coefficients for the partitions a_2*a_1^n for n >= 0 are the Lah numbers A001286. - Tom Copeland, Aug 06 2016

G.f.: 1 / (1 + Sum_{n > 0} a_n x^n/n!) = 1 / exp(a.x). - Tom Copeland, Oct 18 2016

Let a_1 = 1 + x + B_1 = x + 1/2 and a_n = B_n = (B.)^n, where B_n are the Bernoulli numbers defined by e^(B.t) = t / (e^t-1), then t / e^(a.t) = t / [(x + 1) * t + exp(B.t)] =  (e^t - 1) /[ 1 + (x + 1) (e^t - 1)] = exp(p.(x)t), where (p.(x))^n = p_n(x) are the shifted signed polynomials of A019538: p_0(x) = 0, p_1(x) = 1, p_2(x) = -(1 + 2 x), p_3(x) = 1 + 6 x + 6 x^2, ... , p_n(x) = n * b_{n-1}. - Tom Copeland, Oct 18 2016

With a_n = 1/(n+1), b_n = B_n, the Bernoulli numbers. - Tom Copeland, Nov 08 2016

Indeterminate substitutions as illustrated in A356145 lead to [E] = [L][P] = [P][E]^(-1)[P] = [P][RT] and [E]^(-1) = [P][L] = [P][E][P] = [RT][P], where [E] contains the refined Eulerian partition polynomials of A145271; [E]^(-1), A356145, the inverse set to [E]; [P], the permutahedra polynomials of this entry; [L], the classic Lagrange inversion polynomials of A134685; and [RT], the reciprocal tangent polynomials of A356144. Since [L]^2 = [P]^2 = [RT]^2 = [I], the substitutional identity, [L] = [E][P] = [P][E]^(-1) = [RT][P], [RT] = [E]^(-1)[P] = [P][L][P] = [P][E], and [P] = [L][E] = [E][RT] = [E]^(-1)[L] = [RT][E]^(-1). - Tom Copeland, Oct 05 2022


Table starts:

[0] [ 1]

[1] [-1]

[2] [-1,  2]

[3] [-1,  6, -6]

[4] [-1,  8,  6, -36,  24]

[5] [-1, 10, 20, -60, -90,  240, -120]

[6] [-1, 12, 30, -90,  20, -360,  480, -90, 1080, -1800, 720]


b[0] = 1; b[n_] := b[n] = -Sum[Binomial[n, j]*a[j]*b[n-j], {j, 1, n}];

row[0] = {1}; row[n_] := Coefficient[b[n], #]& /@ (Times @@ (a /@ #)&) /@ IntegerPartitions[n];

Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Apr 23 2014 *)



def A133314_row(n): return [(-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in Partitions(n)]

for n in (0..10): print(A133314_row(n)) # Peter Luschny, Sep 18 2015


Row lengths in A000041.

Cf. A000110, A000129, A000262, A000587, A000670, A001286, A007318, A019538, A049019, A084358, A094587, A105278, A110327, A110330, A128229, A132440, A132681, A132710, A263633.

Cf. A134685, A145271, A356144, A356145.

Sequence in context: A048998 A213615 A049019 * A208909 A229565 A259477

Adjacent sequences:  A133311 A133312 A133313 * A133315 A133316 A133317




Tom Copeland, Oct 18 2007, Oct 29 2007, Nov 16 2007


More terms from Jean-François Alcover, Apr 23 2014



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