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A134264 Triangle read by rows: coefficients T(j,k) of a partition transform for Lagrange compositional inversion of a function or generating series in terms of the coefficients of the power series for its reciprocal. Enumeration of noncrossing partitions. 22

%I

%S 1,1,1,1,1,3,1,1,4,2,6,1,1,5,5,10,10,10,1,1,6,6,3,15,30,5,20,30,15,1,

%T 1,7,7,7,21,42,21,21,35,105,35,35,70,21,1

%N Triangle read by rows: coefficients T(j,k) of a partition transform for Lagrange compositional inversion of a function or generating series in terms of the coefficients of the power series for its reciprocal. Enumeration of noncrossing partitions.

%C Matrix begins

%C 1;

%C 1;

%C 1, 1;

%C 1, 3, 1;

%C 1, 4, 2, 6, 1;

%C 1, 5, 5, 10, 10, 10, 1;

%C 1, 6, 6, 3, 15, 30, 5, 20, 30, 15, 1;

%C 1, 7, 7, 7, 21, 42, 21, 21, 35, 105, 35, 35, 70, 21, 1;

%C ...

%C Given an invertible function f(t) analytic about t=0 (or a formal power series) with f(0)=0 and Df(0) not equal 0, form h(t) = t / f(t) and denote h_n as the coefficient of t^n in h(t).

%C Lagrange inversion gives the compositional inverse about t=0 as g(t) = Sum_{j>=1} ( t^j * (1/j) * Sum_{permutations s with s(1) + s(2) + ... + s(j) = j - 1} h_s(1) * h_s(2) * ... * h_s(j) ) = t * T(1,1) * h_0 + Sum_{j>=2} ( t^j * Sum_{k=1..(# of partitions for j-1)} T(j,k) * H(j-1,k ; h_0,h_1,...) ), where H(j-1,k ; h_0,h_1,...) is the k-th partition for h_1 through h_(j-1) corresponding to n=j-1 on page 831 of Abramowitz and Stegun (ordered as in A&S) with (h_0)^(j-m)=(h_0)^(n+1-m) appended to each partition subsumed under n and m of A&S.

%C Denoting h_n by (n') for brevity, to 8th order in t,

%C g(t) = t * (0')

%C + t^2 * [ (0') (1') ]

%C + t^3 * [ (0')^2 (2') + (0') (1')^2 ]

%C + t^4 * [ (0')^3 (3') + 3 (0')^2 (1') (2') + (0') (1')^3 ]

%C + t^5 * [ (0')^4 (4') + 4 (0')^3 (1') (3') + 2 (0')^3 (2')^2 + 6 (0')^2 (1')^2 (2') + (0') (1')^4 ]

%C + t^6 * [ (0')^5 (5') + 5 (0')^4 (1') (4') + 5 (0')^4 (2') (3') + 10 (0')^3 (1')^2 (3') + 10 (0')^3 (1') (2')^2 + 10 (0')^2 (1')^3 (2') + (0') (1')^5 ]

%C + t^7 * [ (0')^6 (6') + 6 (0')^5 (1') (5') + 6 (0')^5 (2') (4') + 3 (0')^5 (3')^2 + 15 (0')^4 (1')^2 (4') + 30 (0')^4 (1') (2') (3') + 5 (0')^4 (2')^3 + 20 (0')^3 (1')^3 (3') + 30 (0')^3 (1')^2 (2')^2 + 15 (0')^2 (1')^4 (2') + (0') (1')^6]

%C + t^8 * [ (0')^7 (7') + 7 (0')^6 (1') (6') + 7 (0')^6 (2') (5') + 7 (0')^6 (3') (4') + 21 (0')^5 (1')^2* (5') + 42 (0')^5 (1') (2') (4') + 21 (0')^5 (1') (3')^2 + 21 (0')^5 (2')^2 (3') + 35 (0')^4 (1')^3 (4') + 105 (0)^4 (1')^2 (2') (3') + 35 (0')^4 (1') (2')^3 + 35 (0')^3 (1')^4 (3') + 70 (0')^3 (1')^3 (2')^2 + 21 (0')^2 (1')^5 (2') + (0') (1')^7 ]

%C + ... , where from the formula section, for example, T(8,1',2',..,7') = 7! / {[8 - (1'+ 2' + ... + 7')]! * 1'! * 2'! * ... 7'!} are the coefficients of the integer partitions (1')^1' (2')^2' ... (7')^7' in the t^8 term.

%C A125181 is an extended, reordered version of the above sequence, omitting the leading 1, with alternate interpretations.

%C If the coefficients of partitions with the same number of h_0 are summed within rows, A001263 is obtained, omitting the leading 1.

%C From identification of the elements of the inversion with those on page 25 of the Ardila et al. link, the coefficients of the irregular table enumerate non-crossing partitions on [n]. - _Tom Copeland_, Oct 13 2014

%C From _Tom Copeland_, Oct 28-29 2014: (Start)

%C Operating with d/d(1') = d/d(h_1) on the n-th partition polynomial Prt(n;h_0,h_1,..,h_n) in square brackets above associated with t^(n+1) generates n * Prt(n-1;h_0,h_1,..,h_(n-1)); therefore, the polynomials are an Appell sequence of polynomials in the indeterminate h_1 when h_0=1 (a special type of Sheffer sequence).

%C Consequently, umbrally, [Prt(.;1,x,h_2,..) + y]^n = Prt(n;1,x+y,h_2,..); that is, Sum_{k=0..n} binomial(n,k) * Prt(k;1,x,h_2,..) * y^(n-k) = Prt(n;1,x+y,h_2,..).

%C Or, e^(x*z) * exp[Prt(.;1,0,h_2,..) * z] = exp[Prt(.;1,x,h_2,..) * z]. Then with x = h_1 = -(1/2) * d^2[f(t)]/dt^2 evaluated at t=0, the formal Laplace transform from z to 1/t of this expression generates g(t), the comp. inverse of f(t), when h_0 = 1 = df(t)/dt eval. at t=0.

%C I.e., t / (1 - t*(x + Prt(.;1,0,h_2,..))) = t / (1 - t*Prt(.;1,x,h_2,..)) = g(t), interpreted umbrally, when h_0 = 1.

%C (End)

%C Connections to and between arrays associated to the Catalan (A000108 and A007317), Riordan (A005043), Fibonacci (A000045), and Fine (A000957) numbers and to lattice paths, e.g., the Motzkin, Dyck, and Łukasiewicz, can be made explicit by considering the inverse in x of the o.g.f. of A104597(x,-t), i.e., f(x) = P(Cinv(x),t-1) = Cinv(x) / (1 + (t-1)*Cinv(x)) = x*(1-x) / (1 + (t-1)*x*(1-x)) = (x-x^2) / (1 + (t-1)*(x-x^2)), where Cinv(x) = x*(1-x) is the inverse of C(x) = (1 - sqrt(1-4*x)) / 2, a shifted o.g.f. for the Catalan numbers, and P(x,t) = x / (1+t*x) with inverse Pinv(x,t) = -P(-x,t) = x / (1-t*x). Then h(x,t) = x / f(x,t) = x * (1+(t-1)Cinv(x)) / Cinv(x) = 1 + t*x + x^2 + x^3 + ... , i.e., h_1=t and all other coefficients are 1, so the inverse of f(x,t) in x, which is explicitly in closed form finv(x,t) = C(Pinv(x,t-1)), is given by A091867, whose coefficients are sums of the refined Narayana numbers above obtained by setting h_1=(1')=t in the partition polynomials and all other coefficients to one. The group generators C(x) and P(x,t) and their inverses allow associations to be easily made between these classic number arrays. - _Tom Copeland_, Nov 03 2014

%C From _Tom Copeland_, Nov 10 2014: (Start)

%C Inverting in x with t a parameter, let F(x;t,n) = x - t*x^(n+1). Then h(x) = x / F(x;t,n) = 1 / (1-t*x^n) = 1 + t*x^n + t^2*x^(2n) + t^3*x^(3n) + ... , so h_k vanishes unless k = m*n with m an integer in which case h_k = t^m.

%C Finv(x;t,n) = Sum_{j>=0} {binomial((n+1)*j,j) / (n*j + 1)} * t^j * x^(n*j + 1), which gives the Catalan numbers for n=1, and the Fuss-Catalan sequences for n>1 (see A001764, n=2). [Added braces to disambiguate the formula. - _N. J. A. Sloane_, Oct 20 2015]

%C This relation reveals properties of the partitions and sums of the coefficients of the array. For n=1, h_k = t^k for all k, implying that the row sums are the Catalan numbers. For n = 2, h_k for k odd vanishes, implying that there are no blocks with only even indexed h_k on the even numbered rows and that only the blocks containing only even-sized bins contribute to the odd-row sums giving the Fuss-Catalan numbers for n=2. And so on, for n>2.

%C These relations are reflected in any combinatorial structures enumerated by this array and the partitions, such as the noncrossing partitions depicted for a five element set (a pentagon) in Wikipedia.

%C (End)

%C From _Tom Copeland_, Nov 12 2014:(Start)

%C An Appell sequence possesses an umbral inverse sequence (cf. A249548). The partition polynomials here, Prt(n;1,h_1,...), are an Appell sequence in the indeterminate h_1=u, so have an e.g.f. exp[Prt(.;1,u,h_2...)*t] = e^(u*t) * exp[Prt(.;1,0,h2,...)*t] with umbral inverses with an e.g.f e^(-u*t) / exp[Prt(.;1,0,h2,...)*t]. This makes contact with the formalism of A133314 (cf. also A049019 and A019538) and the signed, refined face partition polynomials of the permutahedra (or their duals), which determine the reciprocal of exp[Prt(.,0,u,h2...)*t] (cf. A249548) or exp[Prt(.;1,u,h2,...)*t], forming connections among the combinatorics of permutahedra and the noncrossing partitions, Dyck paths and trees (cf. A125181), and many other important structures isomorphic to the partitions of this entry, as well as to formal cumulants through A127671 and algebraic structures of Lie algebras. (Cf. relationship of permutahedra with the Eulerians A008292.)

%C (End)

%C From _Tom Copeland_, Nov 24 2014: (Start)

%C The n-th row multiplied by n gives the number of terms in the homogeneous symmetric monomials generated by [x(1) + x(2) + ... + x(n+1)]^n under the umbral mapping x(m)^j = h_j, for any m. E.g., [a + b + c]^2 = [a^2 + b^2 + c^2] + 2 * [a*b + a*c + b*c] is mapped to [3 * h_2] + 2 * [3 * h_1^2], and 3 * A134264(3) = 3 *(1,1)= (3,3) the number of summands in the two homogeneous polynomials in the square brackets. For n=3, [a + b + c + d]^3 = [a^3 + b^3 + ...] + 3 [a*b^2 + a*c^2 + ...] + 6 [a*b*c + a*c*d + ...] maps to [4 * h_3] + 3 [12 * h_1 * h_2] + 6 [4 * (h_1)^3], and the number of terms in the brackets is given by 4 * A134264(4) = 4 * (1,3,1) = (4,12,4).

%C The further reduced expression is 4 h_3 + 36 h_1 h_2 + 24 (h_1)^3 = A248120(4) with h_0 = 1. The general relation is n * A134264(n) = A248120(n) / A036038(n-1) where the arithmetic is performed on the coefficients of matching partitions in each row n.

%C Abramowitz and Stegun give combinatorial interpretations of A036038 and relations to other number arrays.

%C This can also be related to repeated umbral composition of Appell sequences and topology with the Bernoulli numbers playing a special role. See the Todd class link.

%C (End)

%C These partition polynomials are dubbed the Voiculescu polynomials on page 11 of the He and Jejjala link. - _Tom Copeland_, Jan 16 2015

%C See page 5 of the Josuat-Verges et al. reference for a refinement of these partition polynomials into a noncommutative version composed of nondecreasing parking functions. - _Tom Copeland_, Oct 05 2016

%D A. Nica and R. Speicher (editors), Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series: 335, Cambridge University Press, 2006 (see in particular, Eqn. 9.14 on p. 141, enumerating noncrossing partitions).

%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 A. Alexandrov, <a href="http://arxiv.org/abs/1404.3402">Enumerative geometry, tau-functions, and Heisenberg-Virasoro algebra</a>, arXiv:hep-th/1404.3402v3, 2015 (p.22).

%H F. Ardila, F. Rincon, L. Williams, <a href="http://arxiv.org/abs/1308.2698">Positroids and non-crossing partitions</a>, arXiv preprint arXiv:1308.2698v2 [math.CO], 2013 (p. 25).

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2014/11/02/compositional-inversion-and-appell-sequences/">Compositional inversion and Appell sequences</a>, Nov 2, 2014.

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2014/12/14/the-hirzebruch-criterion-fo-the-todd-class/">The Hirzebruch criterion for the Todd class</a> Dec. 14, 2014.

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2014/12/23/appell-ops-cumulants-noncrossing-partitions-dyck-lattice-paths-and-inversion/">Appell polynomials, cumulants, noncrossing partitions, Dyck lattice paths, and inversion</a> Dec. 23, 2014.

%H Tom 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 K. Ebrahimi-Fard and F. Patras, <a href="https://arxiv.org/abs/1409.5664">Cumulants, free cumulants and half-shuffles</a>, arXiv:1409.5664v2 [math.CO], 2015, p. 12.

%H K. Ebrahimi-Fard and F. Patras, <a href="https://arxiv.org/abs/1502.02748">The splitting process in free probability theory</a>, arXiv:1502.02748 [math.CO], 2015, p. 3.

%H Y. He and V. Jejjala, <a href="http://arxiv.org/abs/hep-th/0307293">Modular Matrix Models</a>, arXiv:hep-th/0307293, 2003.

%H M. Josuat-Verges, F. Menous, J. Novelli, and J. Thibon, <a href="http://arxiv.org/abs/1604.04759">Noncommutative free cumulants</a>, arxiv.org/abs/1604.04759 [math.CO], 2016.

%H M. Mastnak and A. Nica, <a href="https://arxiv.org/abs/0807.4169">Hopf algebras and the logarithm of the S-transform in free probability</a>, arXiv:0807.4169v2 [math.OA], p. 28, 2009.

%H J. McCammond, <a href="http://www.math.ucsb.edu/~jon.mccammond/papers/nc-survey-official.pdf">Non-crossing Partitions in Surprising Locations</a>, American Mathematical Monthly 113 (2006) 598-610.

%H M. Mendez, <a href="https://arxiv.org/abs/1610.03602">Combinatorial differential operators in: Faà di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees</a>, arXiv:1610.03602 [math.CO], p. 33-34 Example 10, 2016.

%H A. Schuetz and G. Whieldon, <a href="http://arxiv.org/abs/1401.7194">Polygonal Dissections and Reversions of Series</a>, arxiv: 1401.7194 [math.CO], 2014.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Noncrossing_partition">Noncrossing partition</a>

%H <a href="/index/Lu#Lukasiewicz">Index entries for sequences related to Łukasiewicz</a>

%F For j>1, there are P(j,m;a...) = j! / [ (j-m)! (a_1)! (a_2)! ... (a_(j-1))! ] permutations of h_0 through h_(j-1) in which h_0 is repeated (j-m) times; h_1, repeated a_1 times; and so on with a_1 + a_2 + ... + a_(j-1) = m.

%F If, in addition, a_1 + 2 * a_2 + ... + (j-1) * a_(j-1) = j-1, then each distinct combination of these arrangements is correlated with a partition of j-1.

%F T(j,k) is [ P(j,m;a...) / j ] for the k-th partition of j-1 as described in the comments.

%F For example from g(t) above, T(5,4) = [ 5! / ((5-3)! * 2!) ] / 5 = 6 for the 4th partition under n=5-1=4 with m=3 parts in A&S.

%F From _Tom Copeland_, Sep 30 2011: (Start)

%F Let W(x) = 1/(df(x)/dx)= 1/{d[x/h(x)]/dx}

%F = [(h_0)-1+:1/(1-h.*x):]^2 / {(h_0)-:[h.x/(1-h.x)]^2:}

%F = [(h_0)+(h_1)x+(h_2)x^2+...]^2 / [(h_0)-(h_2)x^2-2(h_3)x^3-3(h_4)x^4-...], where :" ": denotes umbral evaluation of the expression within the colons and h. is an umbral coefficient.

%F Then for the partition polynomials of A134264,

%F Poly[n;h_0,...,h_(n-1)]=(1/n!)(W(x)*d/dx)^n x, evaluated at x=0, and the compositional inverse of f(t) is g(t) = exp(t*W(x)*d/dx) x, evaluated at x=0. Also, dg(t)/dt = W(g(t)), and g(t) gives A001263 with (h_0)=u and (h_n)=1 for n>0 and A000108 with u=1.

%F (End)

%F From _Tom Copeland_, Oct 20 2011: (Start)

%F With exp[x* PS(.,t)] = exp[t*g(x)]=exp[x*W(y)d/dy] exp(t*y) eval. at y=0, the raising (creation) and lowering (annihilation) operators defined by R PS(n,t) = PS(n+1,t) and L PS(n,t) = n*PS(n-1,t) are

%F R = t*W(d/dt) = t*[(h_0)+(h_1)d/dt+(h_2)(d/dt)^2+...]^2 / [(h_0)-(h_2)(d/dt)^2-2(h_3)(d/dt)^3-3(h_4)(d/dt)^4+...], and

%F L =(d/dt)/h(d/dt)=(d/dt) 1/[(h_0)+(h_1)*d/dt+(h_2)*(d/dt)^2+...]

%F Then P(n,t) = (t^n/n!) dPS(n,z)/dz eval. at z=0 are the row polynomials of A134264. (Cf. A139605, A145271, and link therein to Mathemagical Forests for relation to planted trees on p. 13.)

%F (End)

%F Using the formalism of A263634, the raising operator for the partition polynomials of this array with h_0 = 1 begins as R = h_1 + h_2 D + h_3 D^2/2! + (h_4 - h_2^2 ) D^3/3! + (h_5 - 5 h_2 h_3) D^4/4! + (h_6 + 5 h_2^3 - 7 h_3^2 - 9 h_2 h_4) D^5/5! + (h_7 - 14 h_2 h_5 + 56 h_2^2 h_3) D^6/6! + ... with D = d/d(h_1). - _Tom Copeland_, Sep 09 2016

%F Let h(x) = x/f^{-1}(x) = 1/[1-(c_2*x+c_3*x^2+...)], with c_n all greater than zero. Then h_n are all greater than zero and h_0 = 1. Determine P_n(t) from exp[t*f^{-1}(x)] = exp[x*P.(t)] with f^{-1}(x) = x/h(x) expressed in terms of the h_n (cf. A133314 and A263633). Then P_n(b.) = 0 gives a recursion relation for the inversion polynomials of this entry a_n = b_n/n! in terms of the lower order inversion polynomials and P_j(b.)P_k(b.) = P_j(t)P_k(t)|_{t^n = b_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)]. - _Tom Copeland_, Feb 09 2018

%F A raising operator for the partition polynomials with h_0 = 1 regarded as a Sheffer Appell sequence in h_1 is described in A249548. - _Tom Copeland_, Jul 03 2018

%e 1) With f(t) = t / (t-1), then h(t) = -(1-t), giving h_0 = -1, h_1 = 1 and h_n = 0 for n>1. Then g(t) = -t - t^2 - t^3 - ... = t / (t-1).

%e 2) With f(t) = t*(1-t), then h(t) = 1 / (1-t), giving h_n = 1 for all n. The compositional inverse of this f(t) is g(t) = t*A(t) where A(t) is the o.g.f. for the Catalan numbers; therefore the sum over k of T(j,k), i.e., the row sum, is the Catalan number A000108(j-1).

%e 3) With f(t) = (e^(-a*t)-1) / (-a), h(t) = Sum_{n>=0} Bernoulli(n) * (-a*t)^n / n! and g(t) = log(1-a*t) / (-a) = Sum_{n>=1} a^(n-1) * t^n / n. Therefore with h_n = Bernoulli(n) * (-a)^n / n!, Sum_{permutations s with s(1)+s(2)+...+s(j)=j-1} h_s(1) * h_s(2) * ... * h_s(j) = j * Sum_{k=1..(# of partitions for j-1)} T(j,k) * H(j-1,k ; h_0,h_1,...) = a^(j-1). Note, in turn, Sum_{a=1..m} a^(j-1) = (Bernoulli(j,m+1) - Bernoulli(j)) / j for the Bernoulli polynomials and numbers, for j>1.

%e 4) With f(t,x) = t / (x-1+1/(1-t)), then h(t,x) = x-1+1/(1-t), giving (h_0)=x and (h_n)=1 for n>1. Then g(t,x) = (1-(1-x)*t-sqrt(1-2*(1+x)*t+((x-1)*t)^2)) / 2, a shifted o.g.f. in t for the Narayana polynomials in x of A001263.

%e 5) With h(t)= o.g.f. of A075834, but with A075834(1)=2 rather than 1, which is the o.g.f. for the number of connected positroids on [n] (cf. Ardila et al., p. 25), g(t) is the o.g.f. for A000522, which is the o.g.f. for the number of positroids on [n]. (Added Oct 13 2014 by author.)

%e 6) With f(t,x) = x / ((1-t*x)*(1-(1+t)*x)), an o.g.f. for A074909, the reverse face polynomials of the simplices, h(t,x) = (1-t*x) * (1-(1+t)*x) with h_0=1, h_1=-(1+2*t), and h_2=t*(1+t), giving as the inverse in x about 0 the o.g.f. (1+(1+2*t)*x-sqrt(1+(1+2*t)*2*x+x^2)) / (2*t*(1+t)*x) for signed A033282, the reverse face polynomials of the Stasheff polytopes, or associahedra. Cf. A248727. (Added Jan 21 2015 by author.)

%e 7) With f(x,t) = x / ((1+x)*(1+t*x)), an o.g.f. for the polynomials (-1)^n * (1 + t + ... + t^n), h(t,x) = (1+x) * (1+t*x) with h_0=1, h_1=(1+t), and h_2=t, giving as the inverse in x about 0 the o.g.f. (1-(1+t)*x-sqrt(1-2*(1+t)*x+((t-1)*x)^2)) / (2*x*t) for the Narayana polynomials A001263. Cf. A046802. (Added Jan 24 2015 by author.)

%Y (A001263,A119900) = (reduced array, associated g(x)). See A145271 for meaning and other examples of reduced and associated.

%Y Cf. A119900 (e.g.f. for reduced W(x) with (h_0)=t and (h_n)=1 for n>0).

%Y Cf. A248927 and A248120, "scaled" versions of this Lagrange inversion.

%Y Cf. A091867 and A125181, for relations to lattice paths and trees.

%Y Cf. A000045, A000108, A000957, A001764, A000522, A005043, A007317, A033282,A036038, A046802, A074909, A075834, A104597, A145271, A248727.

%Y Cf. A249548 for use of Appell properties to generate the polynomials.

%Y Cf. A133314, A049019, A019538, A127671, and A008292 for relations to permutahedra, Eulerians.

%Y Cf. A263634.

%Y Cf. A263633.

%K nonn,tabf,more

%O 1,6

%A _Tom Copeland_, Jan 14 2008

%E Added explicit t^6, t^7, and t^8 polynomials and extended a(n) terms to include the coefficients of t^8. - _Tom Copeland_, Sep 14 2016

%E Title modified by _Tom Copeland_, May 28 2018

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