A212358 Coefficients of the cycle index polynomial for the alternating group A_n multiplied by n!/2, n>=1, read as partition polynomial.

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

Offset: 1

Author

Wolfdieter Lang, Jun 12 2012

Keywords

• nonn
• tabf

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.

Examples

			Triangle begins:
  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.
		

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 36, (2.2.6).

Links

• Wolfdieter Lang, Rows n=1..10, Z(A_n) for n=1..13.
• Eric Weisstein's World of Mathematics, Alternating Group.

Crossrefs

Cf. A036039 or A102189 for Z(S_n), A212355 for Z(D_n), and A212357 for Z(C_n).

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).