A248120 Triangle read by rows: Lagrange (compositional) inversion of a function in terms of the coefficients of the Taylor series expansion of its reciprocal, scaled version of A248927, n >= 1, k = 1..A000041(n-1).
1, 2, 6, 3, 24, 36, 4, 120, 360, 60, 80, 5, 720, 3600, 1800, 1200, 300, 150, 6, 5040, 37800, 37800, 16800, 3150, 12600, 3150, 420, 630, 252, 7, 40320, 423360, 705600, 235200, 176400, 352800, 58800, 35280, 23520, 35280, 7056, 1960, 1176, 392, 8
Offset: 1
Examples
Triangle begins
1;
2;
6, 3;
24, 36, 4;
120, 360, 60, 80, 5;
720, 3600, 1800, 1200, 300, 150, 6;
5040, 37800, 37800, 16800, 3150, 12600, 3150, 420, 630, 252, 7;
...
For f(t)= e^t-1, h(t)= t/f(t)= t/(e^t-1), the e.g.f. for the Bernoulli numbers, and plugging the Bernoulli numbers into the Lagrange inversion formula gives g(t)= t - t^2/2 + t^3/3 + ... = log(1+t).
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..2087 (rows 1..20)
- 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].
- Tom Copeland, The Hirzebruch criterion for the Todd class, Dec 14 2014.
- A. Scott and A. Sokal, Some variants of the exponential formula, with application to the multivariate Tutte polynomial (alias Potts model), arXiv:0803.1477 [math.CO], 2009.
Crossrefs
Programs
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PARI
C(v)={my(n=vecsum(v), S=Set(v)); (n+1)*n!^2/((n-#v+1)!*prod(i=1, #S, my(x=S[i], c=#select(y->y==x, v)); x!^c*c!))} row(n)=[C(Vec(p)) | p<-Vecrev(partitions(n-1))] { for(n=1, 7, print(row(n))) } \\ Andrew Howroyd, Feb 02 2022
Formula
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.
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.
T(j,k) is (j-1)! P(j,m;a...) / [(2!)^a_2 (3!)^a_3 ... ((j-1)!)^a_(j-1) ] for the k-th partition of j-1. The partitions are in reverse order--from bottom to top--from the order in Abramowitz and Stegun (page 831).
For example, from g(t) above, T(6,3) = 5! * [6!/(3!*2!)]/(2!)^2 = 1800 for the 3rd partition from the bottom under n=6-1=5 with m=3 parts, and T(6,5) = 5! * [6!/4!]/(2!*3!) = 300.
If the initial factorial and final denominator of T(n,k) are removed and the expression divided by j and the partitions reversed in order, then A134264 is obtained, a refinement of the Narayana numbers.
For f(t) = t*e^(-t), g(t) = T(t), the Tree function, which is the e.g.f. of A000169, and h(t) = t/f(t) = e^t, so h_n = 1 for all n in this case; therefore, the row sums are A000169(n) = n^(n-1) = n* A000272(n).
Let W(x) = 1/(df(x)/dx)= 1/{d[x/h(x)]/dx}=1/[d{x/[h_0+h_1*x+ ...]/dx]. Then the partition polynomials above are given by (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)). See A145271.
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 R = t * W(d/dt) and L =(d/dt)/h(d/dt)=(d/dt) 1/[(h_0)+(h_1)*d/dt+(h_2)*(d/dt)^2/2!+...], which will give a lowering operator associated to the refined f-vectors of permutohedra (cf. A133314 and A049019).
Then [dPS(n,z)/dz]/n eval. at z=0 are the row partition polynomials of this entry. (Cf. A139605, A145271, and link therein to Mathemagical Forests for relation to planted trees on p. 13.)
Following the notes connected to the Lagrange reversion theorem in A248927, a generator for the n-th partition polynomial P_n of this entry is (d/dx)^(n-1) (h (x))^n, and -log(1-t*P.) = (t*Q.) / (1 - t*Q.), umbrally, where (Q.)^n = Q_n is the n-th partition polynomial of A248927. - Tom Copeland, Nov 25 2016
With h_0 = 1, the n-th partition polynomial is obtained as the n-th element (with initial index 0) of the first column of M^{n+1}, where M is the matrix with M_{i,j}= binomial(i,j) h_{i-j}, i.e., the lower triangular Pascal matrix with its n-th diagonal multiplied by h_n. This follows from the Lagrange inversion theorem and the relation between powers of matrices such as M and powers of formal Taylor series discussed in A133314. This is equivalent to repeated binomial convolution of the coefficients of the Taylor series with itself. - Tom Copeland, Nov 13 2019
T(n,k) = n*A248927(n,k). - Andrew Howroyd, Feb 02 2022
Extensions
Terms a(31) and beyond from Andrew Howroyd, Feb 02 2022
Comments