CoCalc Public FilesFormal group law positivity.ipynbOpen with one click!
Author: Jair Taylor
Views : 158
Description: Checking a conjectured positivity criterion for formal group laws

Checking positivity for formal group laws

Define a power series f(x)f(x) by 1/(f1)(x)=ϕ(x)1/(f^{-1})'(x) = \phi(x) where ϕ(x)=1+i=1tixii!\phi(x) = 1 + \sum_{i=1}^\infty t_i \frac{x^i}{i!} where tit_i are indeterminate. We conjecture that the monomial coefficients of

f(f1(x1)+f1(x2)+)f(f^{-1}(x_1) + f^{-1}(x_2) + \cdots)

are polynomials with nonnegative coefficients in tit_i. To find these coefficients quickly, we use the fact that [x1n1xknk]f(f1(x1)+f1(x2)+)=Φ(pn1(t)pnk(t))[x_1^{n_1} \cdots x_k^{n_k}] f(f^{-1}(x_1) + f^{-1}(x_2) + \cdots) = \Phi(p_{n_1}(t) \cdots p_{n_k}(t))

where pk(t)p_k(t) are the polynomials defined by etf1(x)=k=0pk(t)xne^{tf^{-1}(x)} = \sum_{k=0}^\infty p_k(t) x^n and Φ\Phi is the linear functional on polynomials in tt defined by Φ(tn)=n![xn]f(x).\Phi(t^n) = n![x^n]f(x). This fact is Theorem 7.3 in my thesis: https://digital.lib.washington.edu/researchworks/handle/1773/36757

In [1]:
max_degree = 8 ### Degree up to which we will compute the coefficients. t = var('t') R.<x> = PowerSeriesRing(SR) svars = [var('t_' + str(i)) for i in range(max_degree+1)] svars[0] = 1 phi = sum([svars[i] * x^i / factorial(i) for i in range(max_degree+1)]) + O(x^(max_degree+1)) finv = phi.inverse().integral() f = finv.reverse() f_coefficients = [a.expand() for a in f.list()] print "Coefficients of f:" for i in range(max_degree+1): print '(1/%d!) (%s)' % (i, str(factorial(i) * f_coefficients[i])) associated_polynomials = (t*finv).exp().list() associated_polynomials = [p.expand() for p in associated_polynomials] print "\nPolynomials p_k:" for i in range(max_degree+1): print '(1/%d!) (%s)' % (i, str(factorial(i) * associated_polynomials[i]))
Coefficients of f: (1/0!) (0) (1/1!) (1) (1/2!) (t_1) (1/3!) (t_1^2 + t_2) (1/4!) (t_1^3 + 4*t_1*t_2 + t_3) (1/5!) (t_1^4 + 11*t_1^2*t_2 + 4*t_2^2 + 7*t_1*t_3 + t_4) (1/6!) (t_1^5 + 26*t_1^3*t_2 + 34*t_1*t_2^2 + 32*t_1^2*t_3 + 15*t_2*t_3 + 11*t_1*t_4 + t_5) (1/7!) (t_1^6 + 57*t_1^4*t_2 + 180*t_1^2*t_2^2 + 122*t_1^3*t_3 + 34*t_2^3 + 192*t_1*t_2*t_3 + 76*t_1^2*t_4 + 15*t_3^2 + 26*t_2*t_4 + 16*t_1*t_5 + t_6) (1/8!) (t_1^7 + 120*t_1^5*t_2 + 768*t_1^3*t_2^2 + 423*t_1^4*t_3 + 496*t_1*t_2^3 + 1494*t_1^2*t_2*t_3 + 426*t_1^3*t_4 + 294*t_2^2*t_3 + 267*t_1*t_3^2 + 474*t_1*t_2*t_4 + 156*t_1^2*t_5 + 56*t_3*t_4 + 42*t_2*t_5 + 22*t_1*t_6 + t_7) Polynomials p_k: (1/0!) (1) (1/1!) (t) (1/2!) (t^2 - t*t_1) (1/3!) (t^3 - 3*t^2*t_1 + 2*t*t_1^2 - t*t_2) (1/4!) (t^4 - 6*t^3*t_1 + 11*t^2*t_1^2 - 6*t*t_1^3 - 4*t^2*t_2 + 6*t*t_1*t_2 - t*t_3) (1/5!) (t^5 - 10*t^4*t_1 + 35*t^3*t_1^2 - 50*t^2*t_1^3 + 24*t*t_1^4 - 10*t^3*t_2 + 40*t^2*t_1*t_2 - 36*t*t_1^2*t_2 + 6*t*t_2^2 - 5*t^2*t_3 + 8*t*t_1*t_3 - t*t_4) (1/6!) (t^6 - 15*t^5*t_1 + 85*t^4*t_1^2 - 225*t^3*t_1^3 + 274*t^2*t_1^4 - 120*t*t_1^5 - 20*t^4*t_2 + 150*t^3*t_1*t_2 - 346*t^2*t_1^2*t_2 + 240*t*t_1^3*t_2 + 46*t^2*t_2^2 - 90*t*t_1*t_2^2 - 15*t^3*t_3 + 63*t^2*t_1*t_3 - 60*t*t_1^2*t_3 + 20*t*t_2*t_3 - 6*t^2*t_4 + 10*t*t_1*t_4 - t*t_5) (1/7!) (t^7 - 21*t^6*t_1 + 175*t^5*t_1^2 - 735*t^4*t_1^3 + 1624*t^3*t_1^4 - 1764*t^2*t_1^5 + 720*t*t_1^6 - 35*t^5*t_2 + 420*t^4*t_1*t_2 - 1771*t^3*t_1^2*t_2 + 3066*t^2*t_1^3*t_2 - 1800*t*t_1^4*t_2 + 196*t^3*t_2^2 - 966*t^2*t_1*t_2^2 + 1080*t*t_1^2*t_2^2 - 35*t^4*t_3 + 273*t^3*t_1*t_3 - 658*t^2*t_1^2*t_3 + 480*t*t_1^3*t_3 - 90*t*t_2^3 + 175*t^2*t_2*t_3 - 360*t*t_1*t_2*t_3 - 21*t^3*t_4 + 91*t^2*t_1*t_4 - 90*t*t_1^2*t_4 + 20*t*t_3^2 + 30*t*t_2*t_4 - 7*t^2*t_5 + 12*t*t_1*t_5 - t*t_6) (1/8!) (t^8 - 28*t^7*t_1 + 322*t^6*t_1^2 - 1960*t^5*t_1^3 + 6769*t^4*t_1^4 - 13132*t^3*t_1^5 + 13068*t^2*t_1^6 - 5040*t*t_1^7 - 56*t^6*t_2 + 980*t^5*t_1*t_2 - 6496*t^4*t_1^2*t_2 + 20188*t^3*t_1^3*t_2 - 29016*t^2*t_1^4*t_2 + 15120*t*t_1^5*t_2 + 616*t^4*t_2^2 - 5488*t^3*t_1*t_2^2 + 15108*t^2*t_1^2*t_2^2 - 12600*t*t_1^3*t_2^2 - 70*t^5*t_3 + 868*t^4*t_1*t_3 - 3794*t^3*t_1^2*t_3 + 6836*t^2*t_1^3*t_3 - 4200*t*t_1^4*t_3 - 1056*t^2*t_2^3 + 2520*t*t_1*t_2^3 + 840*t^3*t_2*t_3 - 4308*t^2*t_1*t_2*t_3 + 5040*t*t_1^2*t_2*t_3 - 56*t^4*t_4 + 448*t^3*t_1*t_4 - 1112*t^2*t_1^2*t_4 + 840*t*t_1^3*t_4 - 630*t*t_2^2*t_3 + 195*t^2*t_3^2 - 420*t*t_1*t_3^2 + 296*t^2*t_2*t_4 - 630*t*t_1*t_2*t_4 - 28*t^3*t_5 + 124*t^2*t_1*t_5 - 126*t*t_1^2*t_5 + 70*t*t_3*t_4 + 42*t*t_2*t_5 - 8*t^2*t_6 + 14*t*t_1*t_6 - t*t_7)
In [2]:
def extract_fgl_coefficient(exponent_tuple, f_coefficients, polynomials): p = SR(1) for i in exponent_tuple: p *= polynomials[i] p = p.expand() return sum([p.coefficient(t,n) * f_coefficients[n] * factorial(n) for n in range(p.degree(t)+1)]).expand()
In [3]:
Sym = SymmetricFunctions(SR) mon = Sym.monomial() print "Monomial coefficients (multiplied by factorials):" F = 0 for n in range(max_degree+1): for P in Partitions(n): coef = extract_fgl_coefficient(P, f_coefficients, associated_polynomials) F += mon(P) * coef print P, coef * factorial(sum(P)) print "\nFGL up to degree %d:" % max_degree print F
Monomial coefficients (multiplied by factorials): [] 0 [1] 1 [2] 0 [1, 1] 2*t_1 [3] 0 [2, 1] 3*t_2 [1, 1, 1] 6*t_1^2 + 6*t_2 [4] 0 [3, 1] 4*t_3 [2, 2] 12*t_1*t_2 + 6*t_3 [2, 1, 1] 36*t_1*t_2 + 12*t_3 [1, 1, 1, 1] 24*t_1^3 + 96*t_1*t_2 + 24*t_3 [5] 0 [4, 1] 5*t_4 [3, 2] 30*t_2^2 + 30*t_1*t_3 + 10*t_4 [3, 1, 1] 60*t_2^2 + 80*t_1*t_3 + 20*t_4 [2, 2, 1] 120*t_1^2*t_2 + 120*t_2^2 + 150*t_1*t_3 + 30*t_4 [2, 1, 1, 1] 420*t_1^2*t_2 + 240*t_2^2 + 360*t_1*t_3 + 60*t_4 [1, 1, 1, 1, 1] 120*t_1^4 + 1320*t_1^2*t_2 + 480*t_2^2 + 840*t_1*t_3 + 120*t_4 [6] 0 [5, 1] 6*t_5 [4, 2] 150*t_2*t_3 + 60*t_1*t_4 + 15*t_5 [4, 1, 1] 300*t_2*t_3 + 150*t_1*t_4 + 30*t_5 [3, 3] 180*t_1*t_2^2 + 60*t_1^2*t_3 + 260*t_2*t_3 + 100*t_1*t_4 + 20*t_5 [3, 2, 1] 900*t_1*t_2^2 + 540*t_1^2*t_3 + 840*t_2*t_3 + 420*t_1*t_4 + 60*t_5 [3, 1, 1, 1] 2160*t_1*t_2^2 + 1560*t_1^2*t_3 + 1680*t_2*t_3 + 960*t_1*t_4 + 120*t_5 [2, 2, 2] 360*t_1^3*t_2 + 1980*t_1*t_2^2 + 1260*t_1^2*t_3 + 1350*t_2*t_3 + 720*t_1*t_4 + 90*t_5 [2, 2, 1, 1] 1440*t_1^3*t_2 + 4680*t_1*t_2^2 + 3420*t_1^2*t_3 + 2700*t_2*t_3 + 1620*t_1*t_4 + 180*t_5 [2, 1, 1, 1, 1] 5400*t_1^3*t_2 + 10800*t_1*t_2^2 + 9000*t_1^2*t_3 + 5400*t_2*t_3 + 3600*t_1*t_4 + 360*t_5 [1, 1, 1, 1, 1, 1] 720*t_1^5 + 18720*t_1^3*t_2 + 24480*t_1*t_2^2 + 23040*t_1^2*t_3 + 10800*t_2*t_3 + 7920*t_1*t_4 + 720*t_5 [7] 0 [6, 1] 7*t_6 [5, 2] 210*t_3^2 + 315*t_2*t_4 + 105*t_1*t_5 + 21*t_6 [5, 1, 1] 420*t_3^2 + 630*t_2*t_4 + 252*t_1*t_5 + 42*t_6 [4, 3] 630*t_2^3 + 1610*t_1*t_2*t_3 + 280*t_1^2*t_4 + 490*t_3^2 + 735*t_2*t_4 + 245*t_1*t_5 + 35*t_6 [4, 2, 1] 1890*t_2^3 + 6930*t_1*t_2*t_3 + 1680*t_1^2*t_4 + 1470*t_3^2 + 2310*t_2*t_4 + 945*t_1*t_5 + 105*t_6 [4, 1, 1, 1] 3780*t_2^3 + 15960*t_1*t_2*t_3 + 4410*t_1^2*t_4 + 2940*t_3^2 + 4620*t_2*t_4 + 2100*t_1*t_5 + 210*t_6 [3, 3, 1] 3780*t_1^2*t_2^2 + 1260*t_1^3*t_3 + 3780*t_2^3 + 13160*t_1*t_2*t_3 + 3220*t_1^2*t_4 + 2100*t_3^2 + 3360*t_2*t_4 + 1400*t_1*t_5 + 140*t_6 [3, 2, 2] 8820*t_1^2*t_2^2 + 3780*t_1^3*t_3 + 6300*t_2^3 + 23520*t_1*t_2*t_3 + 6300*t_1^2*t_4 + 3150*t_3^2 + 5250*t_2*t_4 + 2310*t_1*t_5 + 210*t_6 [3, 2, 1, 1] 23940*t_1^2*t_2^2 + 11340*t_1^3*t_3 + 12600*t_2^3 + 52920*t_1*t_2*t_3 + 15540*t_1^2*t_4 + 6300*t_3^2 + 10500*t_2*t_4 + 5040*t_1*t_5 + 420*t_6 [3, 1, 1, 1, 1] 63000*t_1^2*t_2^2 + 33600*t_1^3*t_3 + 25200*t_2^3 + 117600*t_1*t_2*t_3 + 37800*t_1^2*t_4 + 12600*t_3^2 + 21000*t_2*t_4 + 10920*t_1*t_5 + 840*t_6 [2, 2, 2, 1] 5040*t_1^4*t_2 + 56700*t_1^2*t_2^2 + 28980*t_1^3*t_3 + 21420*t_2^3 + 92610*t_1*t_2*t_3 + 28980*t_1^2*t_4 + 9450*t_3^2 + 16380*t_2*t_4 + 8190*t_1*t_5 + 630*t_6 [2, 2, 1, 1, 1] 20160*t_1^4*t_2 + 146160*t_1^2*t_2^2 + 81900*t_1^3*t_3 + 42840*t_2^3 + 204120*t_1*t_2*t_3 + 69300*t_1^2*t_4 + 18900*t_3^2 + 32760*t_2*t_4 + 17640*t_1*t_5 + 1260*t_6 [2, 1, 1, 1, 1, 1] 78120*t_1^4*t_2 + 367920*t_1^2*t_2^2 + 226800*t_1^3*t_3 + 85680*t_2^3 + 446040*t_1*t_2*t_3 + 163800*t_1^2*t_4 + 37800*t_3^2 + 65520*t_2*t_4 + 37800*t_1*t_5 + 2520*t_6 [1, 1, 1, 1, 1, 1, 1] 5040*t_1^6 + 287280*t_1^4*t_2 + 907200*t_1^2*t_2^2 + 614880*t_1^3*t_3 + 171360*t_2^3 + 967680*t_1*t_2*t_3 + 383040*t_1^2*t_4 + 75600*t_3^2 + 131040*t_2*t_4 + 80640*t_1*t_5 + 5040*t_6 [8] 0 [7, 1] 8*t_7 [6, 2] 980*t_3*t_4 + 588*t_2*t_5 + 168*t_1*t_6 + 28*t_7 [6, 1, 1] 1960*t_3*t_4 + 1176*t_2*t_5 + 392*t_1*t_6 + 56*t_7 [5, 3] 7280*t_2^2*t_3 + 3360*t_1*t_3^2 + 5320*t_1*t_2*t_4 + 840*t_1^2*t_5 + 2800*t_3*t_4 + 1736*t_2*t_5 + 504*t_1*t_6 + 56*t_7 [5, 2, 1] 21840*t_2^2*t_3 + 13440*t_1*t_3^2 + 21000*t_1*t_2*t_4 + 4200*t_1^2*t_5 + 8400*t_3*t_4 + 5376*t_2*t_5 + 1848*t_1*t_6 + 168*t_7 [5, 1, 1, 1] 43680*t_2^2*t_3 + 30240*t_1*t_3^2 + 47040*t_1*t_2*t_4 + 10416*t_1^2*t_5 + 16800*t_3*t_4 + 10752*t_2*t_5 + 4032*t_1*t_6 + 336*t_7 [4, 4] 5040*t_1*t_2^3 + 4480*t_1^2*t_2*t_3 + 560*t_1^3*t_4 + 13300*t_2^2*t_3 + 6020*t_1*t_3^2 + 9380*t_1*t_2*t_4 + 1540*t_1^2*t_5 + 3780*t_3*t_4 + 2380*t_2*t_5 + 700*t_1*t_6 + 70*t_7 [4, 3, 1] 35280*t_1*t_2^3 + 56560*t_1^2*t_2*t_3 + 8960*t_1^3*t_4 + 61600*t_2^2*t_3 + 35840*t_1*t_3^2 + 58520*t_1*t_2*t_4 + 12040*t_1^2*t_5 + 15400*t_3*t_4 + 10360*t_2*t_5 + 3640*t_1*t_6 + 280*t_7 [4, 2, 2] 60480*t_1*t_2^3 + 112560*t_1^2*t_2*t_3 + 20160*t_1^3*t_4 + 96600*t_2^2*t_3 + 59640*t_1*t_3^2 + 99120*t_1*t_2*t_4 + 21840*t_1^2*t_5 + 23100*t_3*t_4 + 15960*t_2*t_5 + 5880*t_1*t_6 + 420*t_7 [4, 2, 1, 1] 136080*t_1*t_2^3 + 280560*t_1^2*t_2*t_3 + 53760*t_1^3*t_4 + 193200*t_2^2*t_3 + 131040*t_1*t_3^2 + 216720*t_1*t_2*t_4 + 51240*t_1^2*t_5 + 46200*t_3*t_4 + 31920*t_2*t_5 + 12600*t_1*t_6 + 840*t_7 [4, 1, 1, 1, 1] 302400*t_1*t_2^3 + 688800*t_1^2*t_2*t_3 + 142800*t_1^3*t_4 + 386400*t_2^2*t_3 + 285600*t_1*t_3^2 + 470400*t_1*t_2*t_4 + 119280*t_1^2*t_5 + 92400*t_3*t_4 + 63840*t_2*t_5 + 26880*t_1*t_6 + 1680*t_7 [3, 3, 2] 30240*t_1^3*t_2^2 + 10080*t_1^4*t_3 + 126000*t_1*t_2^3 + 233520*t_1^2*t_2*t_3 + 43680*t_1^3*t_4 + 148400*t_2^2*t_3 + 90720*t_1*t_3^2 + 155680*t_1*t_2*t_4 + 35280*t_1^2*t_5 + 31360*t_3*t_4 + 22400*t_2*t_5 + 8400*t_1*t_6 + 560*t_7 [3, 3, 1, 1] 90720*t_1^3*t_2^2 + 30240*t_1^4*t_3 + 282240*t_1*t_2^3 + 572320*t_1^2*t_2*t_3 + 113120*t_1^3*t_4 + 296800*t_2^2*t_3 + 198240*t_1*t_3^2 + 338240*t_1*t_2*t_4 + 81760*t_1^2*t_5 + 62720*t_3*t_4 + 44800*t_2*t_5 + 17920*t_1*t_6 + 1120*t_7 [3, 2, 2, 1] 231840*t_1^3*t_2^2 + 90720*t_1^4*t_3 + 504000*t_1*t_2^3 + 1092000*t_1^2*t_2*t_3 + 231840*t_1^3*t_4 + 468720*t_2^2*t_3 + 322560*t_1*t_3^2 + 562800*t_1*t_2*t_4 + 142800*t_1^2*t_5 + 94080*t_3*t_4 + 68880*t_2*t_5 + 28560*t_1*t_6 + 1680*t_7 [3, 2, 1, 1, 1] 655200*t_1^3*t_2^2 + 272160*t_1^4*t_3 + 1108800*t_1*t_2^3 + 2607360*t_1^2*t_2*t_3 + 588000*t_1^3*t_4 + 937440*t_2^2*t_3 + 695520*t_1*t_3^2 + 1209600*t_1*t_2*t_4 + 325920*t_1^2*t_5 + 188160*t_3*t_4 + 137760*t_2*t_5 + 60480*t_1*t_6 + 3360*t_7 [3, 1, 1, 1, 1, 1] 1814400*t_1^3*t_2^2 + 813120*t_1^4*t_3 + 2419200*t_1*t_2^3 + 6155520*t_1^2*t_2*t_3 + 1478400*t_1^3*t_4 + 1874880*t_2^2*t_3 + 1491840*t_1*t_3^2 + 2587200*t_1*t_2*t_4 + 739200*t_1^2*t_5 + 376320*t_3*t_4 + 275520*t_2*t_5 + 127680*t_1*t_6 + 6720*t_7 [2, 2, 2, 2] 20160*t_1^5*t_2 + 594720*t_1^3*t_2^2 + 252000*t_1^4*t_3 + 907200*t_1*t_2^3 + 2056320*t_1^2*t_2*t_3 + 463680*t_1^3*t_4 + 740880*t_2^2*t_3 + 521640*t_1*t_3^2 + 932400*t_1*t_2*t_4 + 246960*t_1^2*t_5 + 141120*t_3*t_4 + 105840*t_2*t_5 + 45360*t_1*t_6 + 2520*t_7 [2, 2, 2, 1, 1] 80640*t_1^5*t_2 + 1643040*t_1^3*t_2^2 + 735840*t_1^4*t_3 + 1985760*t_1*t_2^3 + 4853520*t_1^2*t_2*t_3 + 1159200*t_1^3*t_4 + 1481760*t_2^2*t_3 + 1118880*t_1*t_3^2 + 1995840*t_1*t_2*t_4 + 559440*t_1^2*t_5 + 282240*t_3*t_4 + 211680*t_2*t_5 + 95760*t_1*t_6 + 5040*t_7 [2, 2, 1, 1, 1, 1] 322560*t_1^5*t_2 + 4455360*t_1^3*t_2^2 + 2126880*t_1^4*t_3 + 4314240*t_1*t_2^3 + 11340000*t_1^2*t_2*t_3 + 2872800*t_1^3*t_4 + 2963520*t_2^2*t_3 + 2388960*t_1*t_3^2 + 4253760*t_1*t_2*t_4 + 1260000*t_1^2*t_5 + 564480*t_3*t_4 + 423360*t_2*t_5 + 201600*t_1*t_6 + 10080*t_7 [2, 1, 1, 1, 1, 1, 1] 1270080*t_1^5*t_2 + 11854080*t_1^3*t_2^2 + 6068160*t_1^4*t_3 + 9313920*t_1*t_2^3 + 26248320*t_1^2*t_2*t_3 + 7056000*t_1^3*t_4 + 5927040*t_2^2*t_3 + 5080320*t_1*t_3^2 + 9031680*t_1*t_2*t_4 + 2822400*t_1^2*t_5 + 1128960*t_3*t_4 + 846720*t_2*t_5 + 423360*t_1*t_6 + 20160*t_7 [1, 1, 1, 1, 1, 1, 1, 1] 40320*t_1^7 + 4838400*t_1^5*t_2 + 30965760*t_1^3*t_2^2 + 17055360*t_1^4*t_3 + 19998720*t_1*t_2^3 + 60238080*t_1^2*t_2*t_3 + 17176320*t_1^3*t_4 + 11854080*t_2^2*t_3 + 10765440*t_1*t_3^2 + 19111680*t_1*t_2*t_4 + 6289920*t_1^2*t_5 + 2257920*t_3*t_4 + 1693440*t_2*t_5 + 887040*t_1*t_6 + 40320*t_7 FGL up to degree 8: m[1] + t_1*m[1, 1] + (t_1^2+t_2)*m[1, 1, 1] + (t_1^3+4*t_1*t_2+t_3)*m[1, 1, 1, 1] + (t_1^4+11*t_1^2*t_2+4*t_2^2+7*t_1*t_3+t_4)*m[1, 1, 1, 1, 1] + (t_1^5+26*t_1^3*t_2+34*t_1*t_2^2+32*t_1^2*t_3+15*t_2*t_3+11*t_1*t_4+t_5)*m[1, 1, 1, 1, 1, 1] + (t_1^6+57*t_1^4*t_2+180*t_1^2*t_2^2+122*t_1^3*t_3+34*t_2^3+192*t_1*t_2*t_3+76*t_1^2*t_4+15*t_3^2+26*t_2*t_4+16*t_1*t_5+t_6)*m[1, 1, 1, 1, 1, 1, 1] + (t_1^7+120*t_1^5*t_2+768*t_1^3*t_2^2+423*t_1^4*t_3+496*t_1*t_2^3+1494*t_1^2*t_2*t_3+426*t_1^3*t_4+294*t_2^2*t_3+267*t_1*t_3^2+474*t_1*t_2*t_4+156*t_1^2*t_5+56*t_3*t_4+42*t_2*t_5+22*t_1*t_6+t_7)*m[1, 1, 1, 1, 1, 1, 1, 1] + 1/2*t_2*m[2, 1] + (3/2*t_1*t_2+1/2*t_3)*m[2, 1, 1] + (7/2*t_1^2*t_2+2*t_2^2+3*t_1*t_3+1/2*t_4)*m[2, 1, 1, 1] + (15/2*t_1^3*t_2+15*t_1*t_2^2+25/2*t_1^2*t_3+15/2*t_2*t_3+5*t_1*t_4+1/2*t_5)*m[2, 1, 1, 1, 1] + (31/2*t_1^4*t_2+73*t_1^2*t_2^2+45*t_1^3*t_3+17*t_2^3+177/2*t_1*t_2*t_3+65/2*t_1^2*t_4+15/2*t_3^2+13*t_2*t_4+15/2*t_1*t_5+1/2*t_6)*m[2, 1, 1, 1, 1, 1] + (63/2*t_1^5*t_2+294*t_1^3*t_2^2+301/2*t_1^4*t_3+231*t_1*t_2^3+651*t_1^2*t_2*t_3+175*t_1^3*t_4+147*t_2^2*t_3+126*t_1*t_3^2+224*t_1*t_2*t_4+70*t_1^2*t_5+28*t_3*t_4+21*t_2*t_5+21/2*t_1*t_6+1/2*t_7)*m[2, 1, 1, 1, 1, 1, 1] + (1/2*t_1*t_2+1/4*t_3)*m[2, 2] + (t_1^2*t_2+t_2^2+5/4*t_1*t_3+1/4*t_4)*m[2, 2, 1] + (2*t_1^3*t_2+13/2*t_1*t_2^2+19/4*t_1^2*t_3+15/4*t_2*t_3+9/4*t_1*t_4+1/4*t_5)*m[2, 2, 1, 1] + (4*t_1^4*t_2+29*t_1^2*t_2^2+65/4*t_1^3*t_3+17/2*t_2^3+81/2*t_1*t_2*t_3+55/4*t_1^2*t_4+15/4*t_3^2+13/2*t_2*t_4+7/2*t_1*t_5+1/4*t_6)*m[2, 2, 1, 1, 1] + (8*t_1^5*t_2+221/2*t_1^3*t_2^2+211/4*t_1^4*t_3+107*t_1*t_2^3+1125/4*t_1^2*t_2*t_3+285/4*t_1^3*t_4+147/2*t_2^2*t_3+237/4*t_1*t_3^2+211/2*t_1*t_2*t_4+125/4*t_1^2*t_5+14*t_3*t_4+21/2*t_2*t_5+5*t_1*t_6+1/4*t_7)*m[2, 2, 1, 1, 1, 1] + (1/2*t_1^3*t_2+11/4*t_1*t_2^2+7/4*t_1^2*t_3+15/8*t_2*t_3+t_1*t_4+1/8*t_5)*m[2, 2, 2] + (t_1^4*t_2+45/4*t_1^2*t_2^2+23/4*t_1^3*t_3+17/4*t_2^3+147/8*t_1*t_2*t_3+23/4*t_1^2*t_4+15/8*t_3^2+13/4*t_2*t_4+13/8*t_1*t_5+1/8*t_6)*m[2, 2, 2, 1] + (2*t_1^5*t_2+163/4*t_1^3*t_2^2+73/4*t_1^4*t_3+197/4*t_1*t_2^3+963/8*t_1^2*t_2*t_3+115/4*t_1^3*t_4+147/4*t_2^2*t_3+111/4*t_1*t_3^2+99/2*t_1*t_2*t_4+111/8*t_1^2*t_5+7*t_3*t_4+21/4*t_2*t_5+19/8*t_1*t_6+1/8*t_7)*m[2, 2, 2, 1, 1] + (1/2*t_1^5*t_2+59/4*t_1^3*t_2^2+25/4*t_1^4*t_3+45/2*t_1*t_2^3+51*t_1^2*t_2*t_3+23/2*t_1^3*t_4+147/8*t_2^2*t_3+207/16*t_1*t_3^2+185/8*t_1*t_2*t_4+49/8*t_1^2*t_5+7/2*t_3*t_4+21/8*t_2*t_5+9/8*t_1*t_6+1/16*t_7)*m[2, 2, 2, 2] + 1/6*t_3*m[3, 1] + (1/2*t_2^2+2/3*t_1*t_3+1/6*t_4)*m[3, 1, 1] + (3*t_1*t_2^2+13/6*t_1^2*t_3+7/3*t_2*t_3+4/3*t_1*t_4+1/6*t_5)*m[3, 1, 1, 1] + (25/2*t_1^2*t_2^2+20/3*t_1^3*t_3+5*t_2^3+70/3*t_1*t_2*t_3+15/2*t_1^2*t_4+5/2*t_3^2+25/6*t_2*t_4+13/6*t_1*t_5+1/6*t_6)*m[3, 1, 1, 1, 1] + (45*t_1^3*t_2^2+121/6*t_1^4*t_3+60*t_1*t_2^3+458/3*t_1^2*t_2*t_3+110/3*t_1^3*t_4+93/2*t_2^2*t_3+37*t_1*t_3^2+385/6*t_1*t_2*t_4+55/3*t_1^2*t_5+28/3*t_3*t_4+41/6*t_2*t_5+19/6*t_1*t_6+1/6*t_7)*m[3, 1, 1, 1, 1, 1] + (1/4*t_2^2+1/4*t_1*t_3+1/12*t_4)*m[3, 2] + (5/4*t_1*t_2^2+3/4*t_1^2*t_3+7/6*t_2*t_3+7/12*t_1*t_4+1/12*t_5)*m[3, 2, 1] + (19/4*t_1^2*t_2^2+9/4*t_1^3*t_3+5/2*t_2^3+21/2*t_1*t_2*t_3+37/12*t_1^2*t_4+5/4*t_3^2+25/12*t_2*t_4+t_1*t_5+1/12*t_6)*m[3, 2, 1, 1] + (65/4*t_1^3*t_2^2+27/4*t_1^4*t_3+55/2*t_1*t_2^3+194/3*t_1^2*t_2*t_3+175/12*t_1^3*t_4+93/4*t_2^2*t_3+69/4*t_1*t_3^2+30*t_1*t_2*t_4+97/12*t_1^2*t_5+14/3*t_3*t_4+41/12*t_2*t_5+3/2*t_1*t_6+1/12*t_7)*m[3, 2, 1, 1, 1] + (7/4*t_1^2*t_2^2+3/4*t_1^3*t_3+5/4*t_2^3+14/3*t_1*t_2*t_3+5/4*t_1^2*t_4+5/8*t_3^2+25/24*t_2*t_4+11/24*t_1*t_5+1/24*t_6)*m[3, 2, 2] + (23/4*t_1^3*t_2^2+9/4*t_1^4*t_3+25/2*t_1*t_2^3+325/12*t_1^2*t_2*t_3+23/4*t_1^3*t_4+93/8*t_2^2*t_3+8*t_1*t_3^2+335/24*t_1*t_2*t_4+85/24*t_1^2*t_5+7/3*t_3*t_4+41/24*t_2*t_5+17/24*t_1*t_6+1/24*t_7)*m[3, 2, 2, 1] + (1/4*t_1*t_2^2+1/12*t_1^2*t_3+13/36*t_2*t_3+5/36*t_1*t_4+1/36*t_5)*m[3, 3] + (3/4*t_1^2*t_2^2+1/4*t_1^3*t_3+3/4*t_2^3+47/18*t_1*t_2*t_3+23/36*t_1^2*t_4+5/12*t_3^2+2/3*t_2*t_4+5/18*t_1*t_5+1/36*t_6)*m[3, 3, 1] + (9/4*t_1^3*t_2^2+3/4*t_1^4*t_3+7*t_1*t_2^3+511/36*t_1^2*t_2*t_3+101/36*t_1^3*t_4+265/36*t_2^2*t_3+59/12*t_1*t_3^2+151/18*t_1*t_2*t_4+73/36*t_1^2*t_5+14/9*t_3*t_4+10/9*t_2*t_5+4/9*t_1*t_6+1/36*t_7)*m[3, 3, 1, 1] + (3/4*t_1^3*t_2^2+1/4*t_1^4*t_3+25/8*t_1*t_2^3+139/24*t_1^2*t_2*t_3+13/12*t_1^3*t_4+265/72*t_2^2*t_3+9/4*t_1*t_3^2+139/36*t_1*t_2*t_4+7/8*t_1^2*t_5+7/9*t_3*t_4+5/9*t_2*t_5+5/24*t_1*t_6+1/72*t_7)*m[3, 3, 2] + 1/24*t_4*m[4, 1] + (5/12*t_2*t_3+5/24*t_1*t_4+1/24*t_5)*m[4, 1, 1] + (3/4*t_2^3+19/6*t_1*t_2*t_3+7/8*t_1^2*t_4+7/12*t_3^2+11/12*t_2*t_4+5/12*t_1*t_5+1/24*t_6)*m[4, 1, 1, 1] + (15/2*t_1*t_2^3+205/12*t_1^2*t_2*t_3+85/24*t_1^3*t_4+115/12*t_2^2*t_3+85/12*t_1*t_3^2+35/3*t_1*t_2*t_4+71/24*t_1^2*t_5+55/24*t_3*t_4+19/12*t_2*t_5+2/3*t_1*t_6+1/24*t_7)*m[4, 1, 1, 1, 1] + (5/24*t_2*t_3+1/12*t_1*t_4+1/48*t_5)*m[4, 2] + (3/8*t_2^3+11/8*t_1*t_2*t_3+1/3*t_1^2*t_4+7/24*t_3^2+11/24*t_2*t_4+3/16*t_1*t_5+1/48*t_6)*m[4, 2, 1] + (27/8*t_1*t_2^3+167/24*t_1^2*t_2*t_3+4/3*t_1^3*t_4+115/24*t_2^2*t_3+13/4*t_1*t_3^2+43/8*t_1*t_2*t_4+61/48*t_1^2*t_5+55/48*t_3*t_4+19/24*t_2*t_5+5/16*t_1*t_6+1/48*t_7)*m[4, 2, 1, 1] + (3/2*t_1*t_2^3+67/24*t_1^2*t_2*t_3+1/2*t_1^3*t_4+115/48*t_2^2*t_3+71/48*t_1*t_3^2+59/24*t_1*t_2*t_4+13/24*t_1^2*t_5+55/96*t_3*t_4+19/48*t_2*t_5+7/48*t_1*t_6+1/96*t_7)*m[4, 2, 2] + (1/8*t_2^3+23/72*t_1*t_2*t_3+1/18*t_1^2*t_4+7/72*t_3^2+7/48*t_2*t_4+7/144*t_1*t_5+1/144*t_6)*m[4, 3] + (7/8*t_1*t_2^3+101/72*t_1^2*t_2*t_3+2/9*t_1^3*t_4+55/36*t_2^2*t_3+8/9*t_1*t_3^2+209/144*t_1*t_2*t_4+43/144*t_1^2*t_5+55/144*t_3*t_4+37/144*t_2*t_5+13/144*t_1*t_6+1/144*t_7)*m[4, 3, 1] + (1/8*t_1*t_2^3+1/9*t_1^2*t_2*t_3+1/72*t_1^3*t_4+95/288*t_2^2*t_3+43/288*t_1*t_3^2+67/288*t_1*t_2*t_4+11/288*t_1^2*t_5+3/32*t_3*t_4+17/288*t_2*t_5+5/288*t_1*t_6+1/576*t_7)*m[4, 4] + 1/120*t_5*m[5, 1] + (1/12*t_3^2+1/8*t_2*t_4+1/20*t_1*t_5+1/120*t_6)*m[5, 1, 1] + (13/12*t_2^2*t_3+3/4*t_1*t_3^2+7/6*t_1*t_2*t_4+31/120*t_1^2*t_5+5/12*t_3*t_4+4/15*t_2*t_5+1/10*t_1*t_6+1/120*t_7)*m[5, 1, 1, 1] + (1/24*t_3^2+1/16*t_2*t_4+1/48*t_1*t_5+1/240*t_6)*m[5, 2] + (13/24*t_2^2*t_3+1/3*t_1*t_3^2+25/48*t_1*t_2*t_4+5/48*t_1^2*t_5+5/24*t_3*t_4+2/15*t_2*t_5+11/240*t_1*t_6+1/240*t_7)*m[5, 2, 1] + (13/72*t_2^2*t_3+1/12*t_1*t_3^2+19/144*t_1*t_2*t_4+1/48*t_1^2*t_5+5/72*t_3*t_4+31/720*t_2*t_5+1/80*t_1*t_6+1/720*t_7)*m[5, 3] + 1/720*t_6*m[6, 1] + (7/144*t_3*t_4+7/240*t_2*t_5+7/720*t_1*t_6+1/720*t_7)*m[6, 1, 1] + (7/288*t_3*t_4+7/480*t_2*t_5+1/240*t_1*t_6+1/1440*t_7)*m[6, 2] + 1/5040*t_7*m[7, 1]