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,
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.
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
KEYWORD
nonn,tabf
AUTHOR
Wolfdieter Lang, Jun 12 2012
STATUS
approved