OFFSET
0,9
COMMENTS
f_n has degree n(n-1), so n-th row of table has n(n-1)+1 entries. Each row is palindromic. The sum of the terms in the n-th row is n!. The first n+1 terms of the n-th row are the same as the first n terms of A052847.
The 'major index' maj(p) of a permutation p = a_1 a_2 ... a_n is the sum of all i such that a_i > a_(i+1). f_n(q) = Sum_p q^(maj(p)+maj(p^(-1))), where the sum is over all permutations of {1,2,...,n}.
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; Exercise 4.20.
LINKS
Alois P. Heinz, Rows n = 0..40, flattened
Zhipeng Lu, Symmetric permutation invariants in some tensor products, arXiv:2103.02168 [math.CO], 2021.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
FORMULA
f_n(q) = Sum_{r=1..n} (-1)^(r+1) q^(r(r-1)/2) (q)_(n-1) (q)_n / ((q)_(r) ((q)_(n-r))^2) f_(n-r)(q) for n>=1.
EXAMPLE
f_0 = f_1 = 1, f_2 = 1+q^2, f_3 = 1+q^2+2q^3+q^4+q^6, so sequence begins 1; 1; 1,0,1; 1,0,1,2,1,0,1; ...
MAPLE
b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
add(b(u-j, o+j-1)*x^(-u), j=1..u)+
add(b(u+j-1, o-j)*x^( o), j=1..o)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n, 0)):
seq(T(n), n=0..10); # Alois P. Heinz, Apr 28 2018
MATHEMATICA
qpoch[x_, n_] := Product[1-x*q^i, {i, 0, n-1}]; f[0]=1; f[n_] := f[n]=Together[Sum[ -(-1)^r q^Binomial[r, 2] qpoch[q^(n-r+1), r-1]*qpoch[q^(r+1), n-r]/qpoch[q, n-r] f[n-r], {r, 1, n}]]; Join@@Table[CoefficientList[f[n], q], {n, 0, 7}]
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
STATUS
approved