%I
%S 1,2,1,6,6,1,24,24,12,12,1,120,120,120,60,60,20,1,720,720,720,360,360,
%T 720,120,120,180,30,1,5040,5040,5040,5040,2520,5040,2520,2520,840,
%U 2520,840,210,420,42,1,40320,40320,40320,40320,20160,20160,40320,40320,20160
%N Numbers associated to partitions, used for combinatoric interpretation of Lah triangle numbers A105278.
%C The order of this array is according to the AbramowitzStegun (ASt) ordering of partitions (see A036036).
%C The row lengths sequence is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
%C These numbers are similar to M_0, M_1, M_2, M_3, M_4 given in A111786, A036038, A036039, A036040, A117506, resp.).
%C Combinatorial interpretation: a(n,k) counts the sets of lists (ordered subsets) obtained from partitioning the set {1..n}, with the lengths of the lists given by the kth partition of n in ASt order. E.g., a(5,5) is computed from the number of sets of lists of lengths [1^1,2^2] (5th partition of 5 in ASt order). Hence a(5,5) = binomial(5,2)*binomial(3,2) = 5!/(1!*2!) = 60 from partitioning the numbers 1,2,...,5 into sets of lists of the type {[.],[..],[..]}.
%C This array, called M_3(2), is the k=2 member of a family of partition arrays generalizing A036040 which appears as M_3 = M_3(k=1). S2(2) = A105278 (unsigned Lah number triangle) is related to M_3(2) in the same way as S2(1), the Stirling2 number triangle, is related to M_3(1).  _Wolfdieter Lang_, Oct 19 2007
%C Another combinatorial interpretation: a(n,k) enumerates unordered forests of increasing binary trees which are described by the kth partition of n in the AbramowitzStegun order.  _Wolfdieter Lang_, Oct 19 2007
%C A relation between partition polynomials formed from these "refined Lah numbers" and Lagrange inversion for an o.g.f. is presented in the link "Lagrange a la Lah" along with an e.g.f. and an umbral binary operator tree representation.  _Tom Copeland_, Apr 12 2011
%C With the indeterminates (x_1,x_2,x_3,...) = (t,c_2*t,c_3*t,...) with c_n >0, umbrally P(n,a.) = P(n,t)_{t^n = a_n} = 0 and P(j,a.)P(k,a.) = P(j,t)P(k,t)_{t^n =a_n} = d_{j,k} >= 0 is the coefficient of x^j/j!*y^k/k! in the Taylor series expansion of the formal group law FGL(x,y) = f[f^{1}(x)+f^{1}(y)], where a_n are the inversion partition polynomials for calculating f(x) from the coefficients of the series expansion of f^{1}(x) given in A133437.  _Tom Copeland_, Feb 09 2018
%C Divided by n!, the row partition polynomials are the elementary homogeneous Schur polynomials presented on p. 44 of the Bracci et al. paper.  _Tom Copeland_, Jun 04 2018
%C Also presented (renormalized) as the Schur polynomials on p. 19 of the Konopelchenko and Schief paper with associations to differential operators related to the KP hierarchy.  _Tom Copeland_, Nov 19 2018
%C Through equation 4.8 on p. 26 of the Arbarello reference, these polynomials appear in the Hirota bilinear equations 4.7 related to taufunction solutions of the KP hierarchy.  _Tom Copeland_, Jan 21 2019
%C These partition polynomials appear as Feynman amplitudes in their Bell polynomial guise (put x_n = n!c_n in A036040 for the indeterminates of the Bell polynomials) in Kreimer and Yeats and Balduf (e.g., p. 27).  _Tom Copeland_, Dec 17 2019
%C From _Tom Copeland_, Oct 15 2020: (Start)
%C With a_n = n! * b_n = (n1)! * c_n for n > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
%C A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
%C B) ordinary generating function (o.g.f.), or formal power series: f(x) = 1/(1b.x) = 1 + Sum_{n > 0} b_n * x^n
%C C) logarithmic generating function (l.g.f): f(x) = 1  log(1  c.x) = 1 + Sum_{n > 0} c_n * x^n /n.
%C Expansions of log(f(x)) are given in
%C I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] = e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
%C II) A263916 for the o.g.f.: log[ 1/(1b.x) ] = log[ 1  F.(b_1,b_2,...)x ] = Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
%C Expansions of exp(f(x)1) are given in
%C III) A036040 for an e.g.f: exp[ e^{a.x}  1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
%C IV) A130561 for an o.g.f.: exp[ b.x/(1b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
%C V) A036039 for an l.g.f.: exp[ log(1c.x) ] = e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
%C Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)
%D E. Arbarello, "Sketches of KdV", Contemp. Math. 312 (2002), p. 969.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H P. Balduf, <a href="http://www2.mathematik.huberlin.de/~kreimer/wpcontent/uploads/PaulMaster">The propagator and diffeomorphisms of an interacting field theory</a>, Master's thesis, submitted to the Institut für Physik, MathematischNaturwissenschaftliche Fakultät, HumboldtUniverstität, Berlin, 2018.
%H F. Bracci, M. Contreras, S. DíazMadrigal, and A. Vasil'ev, <a href="https://arxiv.org/abs/1309.6423">Classical and stochastic LöwnerKufarev equations</a>, arXiv:1309.6423 [math.CV], 2013.
%H T. Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generatorsinversionandmatrixbinomialandintegraltransforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>
%H T. Copeland, <a href="http://tcjpn.wordpress.com/2011/04/11/lagrangealalah/">Lagrange a la Lah</a>
%H T. Copeland, <a href="https://tcjpn.wordpress.com/2018/01/23/formalgrouplawsandbinomialsheffersequences/">Formal group laws and binomial Sheffer sequences</a>, 2018.
%H T. Copeland, <a href="https://tcjpn.wordpress.com/2019/09/13/associahedranoncrossingpartitionsandanumbralalgebraofpowerseries/">In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms</a>, 2019.
%H G. Duchamp, <a href="http://mathoverflow.net/questions/214927/importantformulasincombinatorics/215053#215053">Important formulas in combinatorics: The exponential formula</a>, a Mathoverflow answer, 2015
%H T. Ernst, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.63.274">A history of the qcalculus and a new method</a>, 2000.
%H B. Konopelchenko, <a href="https://arxiv.org/abs/0802.3022">Quantum deformations of associative algebras and integrable systems</a>, arXiv:0802.3022 [nlin.SI], 2008.
%H D. Kreimer and K. Yeats, <a href="http://arxiv.org/abs/1610.01837">Diffeomorphisms of quantum fields</a>, arXiv:1610.01837 [mathph], 2016.
%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Lang/lang.html">Combinatorial Interpretation of Generalized Stirling Numbers</a>, J. Int. Seqs. Vol. 12 (2009) 09.3.3.
%H Wolfdieter Lang, <a href="/A130561/a130561.pdf">First 10 rows and more.</a>
%H J. Novak and M. LaCroix, <a href="https://arxiv.org/abs/1205.2097">Three lectures on free probability</a>, arXiv:1205.2097 [math.CO], 2012.
%F a(n,k) = n!/(Product_{j=1..n} e(n,k,j)!) with the exponent e(n,k,j) of j in the kth partition of n in the ASt ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1.
%F From _Tom Copeland_, Sep 18 2011: (Start)
%F Raising and lowering operators are given for the partition polynomials formed from A130561 in the Copeland link in "Lagrange a la Lah Part I" on pp. 2223.
%F An e.g.f. for the partition polynomials is on page 3:
%F exp[t*:c.*x/(1c.*x):] = exp[t*(c_1*x + c_2*x^2 + c_3*x^3 + ...)] where :(...): denotes umbral evaluation of the enclosed expression and c. is an umbral coefficient. (End)
%F From _Tom Copeland_, Sep 07 2016: (Start)
%F The row partition polynomials of this array P(n,x_1,x_2,...,x_n), given in the Lang link, are n! * S(n,x_1,x_2,...,x_n), where S(n,x_1,...,x_n) are the elementary Schur polynomials, for which d/d(x_m) S(n,x_1,...,x_n) = S(nm,x_1,...,x_(nm)) with S(k,...) = 0 for k < 0, so d/d(x_m) P(n,x_1,...,x_n) = (n!/(nm)!) P(nm,x_1,...,x_(nm)), confirming that the row polynomials form an Appell sequence in the indeterminate x_1 with P(0,...) = 1. See p. 127 of the Ernst paper for more on these Schur polynomials.
%F With the e.g.f. exp[t * P(.,x_1,x_2,..)] = exp(t*x_1) * exp(x_2 t^2 + x_3 t^3 + ...), the e.g.f. for the partition polynomials that form the umbral compositional inverse sequence U(n,x_1,...,x_n) in the indeterminate x_1 is exp[t * U(.,x_1,x_2,...)] = exp(t*x_1) exp[(x_2 t^2 + x_3 t^3 + ...)]; therefore, U(n,x_1,x_2,...,x_n) = P(n,x_1,x_2,.,x_n), so umbrally P[n,P(.,x_1,x_2,x_3,...),x_2,x_3,...,x_n] = (x_1)^n = P[n,P(.,x_1,x_2,...),x_2,x_3,...,x_n]. For example, P(1,x_1) = x_1, P2(x_1,x_2) = 2 x_2 + x_1^2, and P(3,x_1,x_2,x_3) = 6 x_3 + 6 x_2 x_1 + x_1^3, then P[3,P(.,x_1,x_2,...),x_2,x_3] = 6 x_3 + 6 x_2 P(1,x_1) + P(3,x_1,x_2,x_3) = 6 x_3 + 6 x_2 x_1 + 6 (x_3) + 6 (x_2) x_1 + x_1^3 = x_1^3.
%F From the Appell formalism, umbrally [P(.,0,x_2,x_3,...) + y]^n = P(n,y,x_2,x_3,...,x_n).
%F The indeterminates of the partition polynomials can also be extracted using the Faber polynomials of A263916 with n * x_n = F(n,S(1,x_1),...,S(n,x_1,...,x_n)) = F(n,P(1,x_1),...,P(n,x_1,...,x_n)/n!). Compare with A263634.
%F Also P(n,x_1,...,x_n) = ST1(n,x_1,2*x_2,...,n*x_n), where ST1(n,...) are the row partition polynomials of A036039.
%F (End)
%e Triangle starts:
%e [ 1];
%e [ 2, 1];
%e [ 6, 6, 1];
%e [ 24, 24, 12, 12, 1];
%e [120, 120, 120, 60, 60, 20, 1];
%e ...
%e a(5,6) = 20 = 5!/(3!*1!) because the 6th partition of 5 in ASt order is [1^3,2^1].
%e a(5,5) = 60 enumerates the unordered [1^1,2^2]forest with 5 vertices (including the three roots) composed of three such increasing binary trees: 5*((binomial(4,2)*2)*(1*2))/2! = 5*12 = 60.
%Y Cf. A105278 (unsigned Lah triangle L(n, m)) obtained by summing the numbers for given part number m.
%Y Cf. A000262 (row sums), identical with row sums of unsigned Lah triangle A105278.
%Y A134133(n, k) = A130561(n, k)/A036040(n, k) (division by the M_3 numbers).  _Wolfdieter Lang_, Oct 12 2007
%Y Cf. A036039, A263634, A263916.
%Y Cf. A096162.
%Y Cf. A133437.
%Y Cf. A127671.
%K nonn,tabf,easy
%O 1,2
%A _Wolfdieter Lang_, Jul 13 2007
