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%I #126 Feb 16 2022 19:07:38
%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; elementary Schur polynomials / functions.
%C The order of this array is according to the Abramowitz-Stegun (A-St) 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, respectively.
%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 k-th partition of n in A-St 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 A-St 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 k-th partition of n in the Abramowitz-Stegun 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 tau-function 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 = (n-1)! * 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/(1-b.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/(1-b.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/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
%C V) A036039 for an l.g.f.: exp[ -log(1-c.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)
%C These partition polynomials are referred to as Schur functions by Segal and Wilson, who present associations with Plucker coordinates, Grassmannians, and the tau functions of the KdV hierarchy. See pages 51 and 61. - _Tom Copeland_, Jan 08 2022
%D E. Arbarello, "Sketches of KdV", Contemp. Math. 312 (2002), p. 9-69.
%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.hu-berlin.de/~kreimer/wp-content/uploads/PaulMaster">The propagator and diffeomorphisms of an interacting field theory</a>, Master's thesis, submitted to the Institut für Physik, Mathematisch-Naturwissenschaftliche Fakultät, Humboldt-Universtität, Berlin, 2018.
%H F. Bracci, M. Contreras, S. Díaz-Madrigal, and A. Vasil'ev, <a href="https://arxiv.org/abs/1309.6423">Classical and stochastic Löwner-Kufarev equations</a>, arXiv:1309.6423 [math.CV], 2013.
%H T. Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>
%H T. Copeland, <a href="http://tcjpn.wordpress.com/2011/04/11/lagrange-a-la-lah/">Lagrange a la Lah</a>
%H T. Copeland, <a href="https://tcjpn.wordpress.com/2018/01/23/formal-group-laws-and-binomial-sheffer-sequences/">Formal group laws and binomial Sheffer sequences</a>, 2018.
%H T. 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 G. Duchamp, <a href="http://mathoverflow.net/questions/214927/important-formulas-in-combinatorics/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 q-calculus 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 [math-ph], 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.
%H G. Segal and G. Wilson, <a href="http://www.numdam.org/article/PMIHES_1985__61__5_0.pdf">Loop groups and equations of KdV type</a>, Publications mathématiques de l’I.H.É.S., tome 61, 1985, p. 5-65.
%F a(n,k) = n!/(Product_{j=1..n} e(n,k,j)!) with the exponent e(n,k,j) of j in the k-th partition of n in the A-St 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. 22-23.
%F An e.g.f. for the partition polynomials is on page 3:
%F exp[t*:c.*x/(1-c.*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(n-m,x_1,...,x_(n-m)) with S(k,...) = 0 for k < 0, so d/d(x_m) P(n,x_1,...,x_n) = (n!/(n-m)!) P(n-m,x_1,...,x_(n-m)), 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 A-St 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
%E Name augmented by _Tom Copeland_, Dec 08 2022
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