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 A212358 Coefficients of the cycle index polynomial for the alternating group A_n multiplied by n!/2, n>=1, read as partition polynomial. 1
 1, 0, 1, 2, 0, 1, 0, 8, 3, 0, 1, 24, 0, 0, 20, 15, 0, 1, 0, 144, 90, 40, 0, 0, 0, 40, 45, 0, 1, 720, 0, 0, 0, 504, 630, 280, 210, 0, 0, 0, 70, 105, 0, 1, 0, 5760, 3360, 2688, 1260, 0, 0, 0, 0, 0, 1344, 2520, 1120, 1680, 105, 0, 0, 0, 112, 210, 0, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS The row lengths sequence is A000041. The partitions are ordered like in Abramowitz-Stegun (for the reference see A036036, where also a link to a work by C. F. Hindenburg from 1779 is found where this order has been used). The row sums are A001710(n-1), n>=1. The cycle index (multivariate polynomial) for the alternating group A_n, called Z(A_n), is   Z(S_n) + Z(S_n;x[1],-x[2],x[3],-x[4],... ), n>=1,   with the cycle index Z(S_n) for the symmetric group S_n, in the variables x[1],...,x[n]. See the Harary and Palmer reference. The coefficients of n!*Z(S_n) are the M_2 numbers of Abramowitz-Stegun, pp. 831-2. See A036039 and A102189, also for the Abramowitz-Stegun reference. REFERENCES F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 36, (2.2.6). LINKS WolframMathWorld: AlternatingGroup Wolfdieter Lang, Rows n=1..10, Z(A_n)) for n=1..13. FORMULA The cycle index polynomial for the alternating group A_n is Z(A_n) = (2*a(n,k)*x[1]^(e[k,1])*x[2]^(e[k,2])*...*x[n]^(e[k,n]))/n!, n>=1, if the k-th partition of n in Abramowitz-Stegun order is 1^(e[k,1]) 2^(e[k,2]) ... n^(e[k,n]), where a part j with vanishing exponent e[k,j] has to be omitted. The n dependence of the exponents has been suppressed. See the comment above for the Z(A_n) formula, and the link for these polynomials for n=1..13. a(n,k) is the coefficient the term of (n!/2)*Z(A_n) corresponding to the k-th partition of n in Abramowitz-Stegun order. a(n,k) = 0 if there is no such term in Z(A_n). EXAMPLE n\k  1    2    3   4   5  6  7   8   9 10 11 ... 1:   1 2:   0    1 3:   2    0    1 4:   0    8    3   0   1 5:  24    0    0  20  15  0  1 6:   0  144   90  40   0  0  0  40  45  0  1 ... See the link for rows n=1..10 and the Z(A_n) polynomials for n=1..13. n=6: Z(A_6) = 2*(144*x[1]*x[5] + 90*x[2]*x[4] + 40*x[3]^2 + 40*x[1]^3*x[3] + 45*x[1]^2*x[2]^2 + 1*x[1]^6)/6!, because the relevant partitions of 6 appear for k=2: 1,5;  k=3: 2,4; k=4: 3^2; k=8: 1^3,3; k=9: 1^2,2^2  and k=11: 1^6. Thus, Z(A_6) = (2/5)*x[1]*x[5] + (1/4)*x[2]*x[4] +  (1/9)*x[3]^2  + (1/9)*x[1]^3*x[3] + (1/8)*x[1]^2*x[2]^2 + (1/360)*x[1]^6. CROSSREFS Cf.  A036039 or A102189 for Z(S_n), A212355 for Z(D_n), and A212357 for Z(C_n). Sequence in context: A059419 A185415 A049218 * A154469 A037273 A285313 Adjacent sequences:  A212355 A212356 A212357 * A212359 A212360 A212361 KEYWORD nonn,tabf AUTHOR Wolfdieter Lang, Jun 12 2012 STATUS approved

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Last modified January 18 06:34 EST 2019. Contains 319269 sequences. (Running on oeis4.)