

A130561


Numbers associated to partitions, used for combinatoric interpretation of Lah triangle numbers A105278.


21



1, 2, 1, 6, 6, 1, 24, 24, 12, 12, 1, 120, 120, 120, 60, 60, 20, 1, 720, 720, 720, 360, 360, 720, 120, 120, 180, 30, 1, 5040, 5040, 5040, 5040, 2520, 5040, 2520, 2520, 840, 2520, 840, 210, 420, 42, 1, 40320, 40320, 40320, 40320, 20160, 20160, 40320, 40320, 20160
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

The order of this array is according to the AbramowitzStegun (ASt) ordering of partitions (see A036036).
The row lengths sequence is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
These numbers are similar to M_0, M_1, M_2, M_3, M_4 given in A111786, A036038, A036039, A036040, A117506, resp.).
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 {[.],[..],[..]}.
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
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
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
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
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
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
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


REFERENCES

E. Arbarello, "Sketches of KdV", Contemp. Math. 312 (2002), p. 969.


LINKS

Table of n, a(n) for n=1..53.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
F. Bracci, M. Contreras, S. DíazMadrigal, and A. Vasil'ev, Classical and stochastic LöwnerKufarev equations, arXiv:1309.6423 [math.CV], 2013.
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
T. Copeland, Lagrange a la Lah
T. Copeland, Formal group laws and binomial Sheffer sequences, 2018.
G. Duchamp, Important formulas in combinatorics: The exponential formula, a Mathoverflow answer, 2015
T. Ernst, A history of the qcalculus and a new method, 2000.
B. Konopelchenko and W. Schief, Quantum deformations of associative algebras and integrable systems, arXiv:0802.3022 [nlin.SI], 2008.
Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.
Wolfdieter Lang, First 10 rows and more.


FORMULA

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.
From Tom Copeland, Sep 18 2011: (Start)
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.
An e.g.f. for the partition polynomials is on page 3:
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)
From Tom Copeland, Sep 07 2016: (Start)
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.
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.
From the Appell formalism, umbrally [P(.,0,x_2,x_3,...) + y]^n = P(n,y,x_2,x_3,...,x_n).
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.
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.
(End)


EXAMPLE

Triangle starts:
[ 1];
[ 2, 1];
[ 6, 6, 1];
[ 24, 24, 12, 12, 1];
[120, 120, 120, 60, 60, 20, 1];
...
a(5,6) = 20 = 5!/(3!*1!) because the 6th partition of 5 in ASt order is [1^3,2^1].
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.


CROSSREFS

Cf. A105278 (unsigned Lah triangle L(n, m)) obtained by summing the numbers for given part number m.
Cf. A000262 (row sums), identical with row sums of unsigned Lah triangle A105278.
A134133(n, k) = A130561(n, k)/A036040(n, k) (division by the M_3 numbers).  Wolfdieter Lang, Oct 12 2007
Cf. A036039, A263634, A263916.
Cf. A096162.
Cf. A133437.
Sequence in context: A110183 A110098 A244888 * A157400 A091599 A048999
Adjacent sequences: A130558 A130559 A130560 * A130562 A130563 A130564


KEYWORD

nonn,tabf,easy


AUTHOR

Wolfdieter Lang, Jul 13 2007


STATUS

approved



