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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.

%I A212358 #12 Apr 10 2025 23:22:20
%S A212358 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,
%T A212358 720,0,0,0,504,630,280,210,0,0,0,70,105,0,1,0,5760,3360,2688,1260,0,0,
%U A212358 0,0,0,1344,2520,1120,1680,105,0,0,0,112,210,0,1
%N A212358 Coefficients of the cycle index polynomial for the alternating group A_n multiplied by n!/2, n>=1, read as partition polynomial.
%C A212358 The row lengths sequence is A000041.
%C A212358 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).
%C A212358 The row sums are A001710(n-1), n>=1.
%C A212358 The cycle index (multivariate polynomial) for the alternating group A_n, called Z(A_n), is
%C A212358   Z(S_n) + Z(S_n;x[1],-x[2],x[3],-x[4],... ), n>=1,
%C A212358   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.
%D A212358 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 36, (2.2.6).
%H A212358 Wolfdieter Lang, <a href="/A212358/a212358.pdf">Rows n=1..10, Z(A_n) for n=1..13.</a>
%H A212358 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingGroup.html">Alternating Group</a>.
%F A212358 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.
%F A212358 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).
%e A212358 Triangle begins:
%e A212358   n\k  1    2   3   4   5  6  7   8   9 10 11 ...
%e A212358   1:   1
%e A212358   2:   0    1
%e A212358   3:   2    0   1
%e A212358   4:   0    8   3   0   1
%e A212358   5:  24    0   0  20  15  0  1
%e A212358   6:   0  144  90  40   0  0  0  40  45  0  1
%e A212358   ...
%e A212358 See the link for rows n=1..10 and the Z(A_n) polynomials for n=1..13.
%e A212358 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.
%Y A212358 Cf. A036039 or A102189 for Z(S_n), A212355 for Z(D_n), and A212357 for Z(C_n).
%K A212358 nonn,tabf
%O A212358 1,4
%A A212358 _Wolfdieter Lang_, Jun 12 2012