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 A133314 Coefficients of list partition transform: reciprocal of an exponential generating function (e.g.f.). 69

%I

%S 1,-1,-1,2,-1,6,-6,-1,8,6,-36,24,-1,10,20,-60,-90,240,-120,-1,12,30,

%T -90,20,-360,480,-90,1080,-1800,720,-1,14,42,-126,70,-630,840,-420,

%U -630,5040,-4200,2520,-12600,15120,-5040,-1,16,56,-168,112,-1008,1344,70

%N Coefficients of list partition transform: reciprocal of an exponential generating function (e.g.f.).

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

%C b(0) = 1

%C b(1) = -a(1)

%C b(2) = -a(2) + 2*a(1)^2

%C b(3) = -a(3) + 6*a(2)a(1) - 6*a(1)^3

%C b(4) = -a(4) + 8*a(3)a(1) + 6*a(2)^2 - 36*a(2)*a(1)^2 + 24*a(1)^4

%C 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).

%C Expressed umbrally,

%C exp(a(.)*x)*exp(b(.)*x) = 1, i.e., (a(.)+b(.))^n = 1 for n=0 and 0 for all other values of n.

%C Expressed recursively,

%C 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.

%C Expressed in matrix form, b(n) form the first column of B = matrix inverse of A .

%C A = Pascal matrix diagonally multipied by a(n), i.e., A(n,k) = binomial(n,k)*a(n-k).

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

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

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

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

%C 4) A000110 and A000587.

%C 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.

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

%C 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,... .

%C 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).

%C If the matrix equation M = exp(T) also holds, then exp[a(.)*T] * exp[b(.)*T] = exp[(a+b)*T] = Identity matrix because (a(.)+b(.))^n = delta(n), the Kronecker delta.

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

%C 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 diagonally multiplying any of the fundamental matrices by a(.), followed by matrix inversion and then extraction of the b(.) factors from the first column (simplest for the Pascal formulas above).

%C Conversely, the inverses of matrices formed by diagonally multiplying the three fundamental matrices by a(.) are given by diagonally multiplying the fundamental matrices by b(.).

%C 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(.)].

%C 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 =

%C 0;

%C t(0), 0;

%C 0, t(1), 0;

%C 0, 0, t(2), 0;

%C 0, 0, 0, t(3), 0;

%C ...

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

%C [ 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), ... ].

%C 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.

%C 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(.), as in the Pascal matrix example above and their inverses by diagonal multiplication by LPT(a) = b.

%C Weighted-mappings interpretation for the top partition equation:

%C 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.

%C 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).

%C E.g., b(3) = [ 3 Pre to 1 Post ] + [ 3 Pre to 2 Post ] + [ 3 Pre to 3 Post ]

%C = [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]

%C = -a(3) + 6 * [-a(2)*-a(1)] + 6 * [-a(1)*-a(1)*-a(1)].

%C 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

%C 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

%C 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

%C 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

%H Vincenzo Librandi, <a href="/A133314/b133314.txt">Table of n, a(n) for n = 0..250</a>

%H M. Aguiar and F. Ardila, <a href="http://math.sfsu.edu/federico/Talks/GPatMSRI.pdf">The algebraic and combinatorial structure of generalized permutahedra</a>, MSRI Summer School July 19, 2017.

%H M. Aguiar and F. Ardila, <a href="https://arxiv.org/abs/1709.07504">Hopf monoids and generalized permutahedra</a>, arXiv:1709.07504 [math.CO], p. 5, 2017.

%H N. Arkani-Hamed, Y. Bai, S. He, and G. Yan, <a href="https://arxiv.org/abs/1711.09102">Scattering forms and the positive geometry of kinematics, color, and the worldsheet </a>, arXiv:1711.09102 [hep-th], 2017.

%H J. Bergström and S. Minabe, <a href="https://arxiv.org/abs/1108.0338">On the cohomology of the Losev-Manin moduli space</a>, arXiv:1108.0338 [math.AG], (cf. p. 1), 2013.

%H Z. Bern, J.Carrasco, M. Chiodaroli, H. Johansson, and R. Roiban , <a href="https://arxiv.org/abs/1909.01358">The Duality Between Color and Kinematics and its Applications</a>, arXiv preprint arXiv:1909.01358 [hep-th], 2019.

%H L. Berry, S. Forcey, M. Ronco, and P. Showers, <a href="https://arxiv.org/abs/1608.08546">Polytopes and Hopf algebras of painted trees: Fan graphs and Stellohedra</a>, arXiv:1608.08546 [math.CO], 2018.

%H Tom Copeland, <a href="http://math.stackexchange.com/questions/1785526/bijective-mapping-between-face-polytopes-of-permutohedra-and-partitions-of-integ">Bijective mapping between face polytopes of permutohedra and partitions of integers</a>, Math StackExchange question, 2016.

%H Tom Copeland, <a href="https://tcjpn.wordpress.com/2019/09/13/associahedra-noncrossing-partitions-and-an-umbral-algebra-of-power-series/">In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms</a>, 2019.

%H Karl-Dieter Crisman, <a href="http://www.math.gordon.edu/~kcrisman/KemenyBordaPermutahedron-final.pdf">The Borda Count, the Kemeny Rule, and the permutahedron</a>, preprint, 2014.

%H Karl-Dieter Crisman, <a href="http://dx.doi.org/10.1090/conm/624">The Borda Count, the Kemeny Rule, and the permutahedron</a>, 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.

%H R. Da Rosa, D. Jensen, and D. Ranganathan, <a href="https://arxiv.org/abs/1411.0537">Toric graph associahedra and compactifications of M_(0,n)</a>, arXiv:1411.0537 [math.AG], 2015.

%H Nick Early, <a href="https://arxiv.org/abs/1810.03246">Honeycomb tessellations and canonical bases for permutohedral blades</a>, arXiv:1810.03246 [math.CO], 2018.

%H S. Forcey, <a href="http://www.math.uakron.edu/~sf34/hedra.htm#index">The Hedra Zoo</a>

%H S. Franco and A. Hasan, <a href="https://arxiv.org/abs/1904.07954">Graded Quivers, Generalized Dimer Models and Toric Geometry </a>, arXiv preprint arXiv:1904.07954 [hep-th], 2019.

%H M. Futaki and K. Ueda, <a href="https://arxiv.org/abs/1001.4858">Tropical coamoeba and torus-equivariant homological mirror symmetry for the projective space </a>, arXiv preprint arXiv:1001.4858 [math.SG], 2010-2014.

%H X. Gao, S. He, Y. Zhan, <a href="https://arxiv.org/abs/1708.08701">Labelled tree graphs, Feynman diagrams and disk integrals </a>, arXiv:1708.08701 [hep-th], 2017.

%H A. Hasan, <a href="https://academicworks.cuny.edu/gc_etds/3321/">Physics and Mathematics of Graded Quivers</a>, dissertation, Graduate Center, City University of New York, 2019.

%H D. Karp, D. Ranganathan, P. Riggins, and U. Whitcher, <a href="https://arxiv.org/abs/1109.5157">Toric Symmetry of CP^3</a>, arXiv:1109.5157 [math.AG], 2011.

%H R. Kaufmann and Y. Zhang, <a href="https://arxiv.org/abs/1602.08247">Permutohedral structures on E2-operads</a>, arXiv preprint arXiv:1602.08247 [math.AT], 2016.

%H M. Lin, <a href="https://www.math.hmc.edu/~mlin/thesis/mlin-2016-thesis.pdf">Graph Cohomology</a>, 2016.

%H J. Loday, <a href="http://www-irma.u-strasbg.fr/~loday/PAPERS/MultFAsENG2.pdf">The Multiple Facets of the Associahedron</a>

%H MathOverflow, <a href="https://mathoverflow.net/questions/270070/analogue-of-conic-sections-for-the-permutohedra-associahedra-and-noncrossing-p">Analogue of conic sections for the permutohedra, associahedra, and noncrossing partitions</a>, an MO question posed by T. Copeland, 2017.

%H W. Norledge and A. Ocneanu, <a href="https://arxiv.org/abs/1911.11736">Hopf monoids, permutohedral cones, and generalized retarded functions</a>, arXiv preprint arXiv:1911.11736 [math.CO], 2020.

%H P. Olver, <a href="http://www.math.umn.edu/~olver/di_/contact.pdf">The canonical contact form</a>, pg. 9.

%H V. Pilaud, <a href="https://www.mat.univie.ac.at/~slc/wpapers/s76vortrag/pilaud.pdf">The Associahedron and its Friends</a>, presentation for Séminaire Lotharingien de Combinatoire, April 4 - 6, 2016.

%H J . Pitman and R. Stanley, <a href="https://arxiv.org/abs/math/9908029">A polytope related to empirical distributions, plane trees, parking functions, and the associahedron</a>, arxiv: 9908029 [math.CO], 1999.

%H A. Postnikov, <a href="https://arxiv.org/abs/1806.05307">Positive Grassmannian and Polyhedral Subdivisions</a>, arXiv:1806.05307 [math.CO], (cf. p. 17), 2018.

%H D. Ranganathan, <a href="https://www.math.hmc.edu/seniorthesis/archives/2012/dranga/">Gromov-Witten Theory of Blowups of Toric Threefolds</a>, senior thesis, Harvey Mudd College, 2012.

%H D. Ranganathan, <a href="https://www.math.hmc.edu/seniorthesis/archives/2012/dranga/">Gromov-Witten Theory of Blowups of Toric Threefolds (poster)</a>, poster for senior thesis, Harvey Mudd College, 2012.

%F 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

%F (E.g.f. of matrix B) = (e.g.f. of b)·exp(x·t) = exp(b(.)·t)·exp(x·t) = exp(x·t)/exp(a(·)·t) = (e.g.f. of A^(-1)) and (e.g.f. of matrix A) = exp(a(·)·t)·exp(x·t) = exp(x·t)/exp(b(·)·t) = (e.g.f. of B^(-1)). These e.g.f. define Appell sequences of polynomials. - _Tom Copeland_, Mar 22 2014

%F Sum of the n-th row is (-1)^n. - _Peter Luschny_, Sep 18 2015

%F The unsigned coefficients for the partitions a(2)*a(1)^n for n >= 0 are the Lah numbers A001286. - _Tom Copeland_, Aug 06 2016

%F G.f.: 1 / (1 + Sum_{n > 0} a(n)*x^n/n!) = 1 / exp(a.x). - _Tom Copeland_, Oct 18 2016

%F 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

%F With a(n) = 1/(n+1), b(n) = B(n), the Bernoulli numbers. - _Tom Copeland_, Nov 08 2016

%e Table starts:

%e  [ 1]

%e  [-1]

%e  [-1, 2]

%e  [-1, 6, -6]

%e  [-1, 8, 6, -36, 24]

%e  [-1, 10, 20, -60, -90, 240, -120]

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

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

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

%t Table[row[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Apr 23 2014 *)

%o (Sage)

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

%o for n in (0..10): print(A133314_row(n)) # _Peter Luschny_, Sep 18 2015

%Y Row lengths in A000041.

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

%K sign,tabf

%O 0,4

%A _Tom Copeland_, Oct 18 2007, Oct 29 2007, Nov 16 2007

%E More terms from _Jean-François Alcover_, Apr 23 2014

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