OFFSET
1,5
COMMENTS
This sequence is from Table 1.1 of Chen and Wang, p. 2. Abstract: We show that the distribution of the coefficients of the q-derangement numbers is asymptotically normal. We also show that this property holds for the q-derangement numbers of type B.
Number of terms in row n appears to be A084265(n+2). - N. J. A. Sloane, Jul 20 2008
T(n,k) is the number of derangements in the set S(n) of permutations of {1,2,...,n} having major index equal to k. Example: T(4,3)=2 because we have 4312 (descent positions 1 and 2) and 2341 (descent position 3). - Emeric Deutsch, May 04 2009
LINKS
William Y. C. Chen and David G. L. Wang, The Limiting Distributions of the Coefficients of the q-Derangement Number, arXiv:0806.2092 [math.CO], 2008.
FORMULA
T(n,k) = [q^k] { [n]_q! * Sum_{m=0..n} (-1)^m*q^(m(m-1)/2) / [m]_q! } for n>=2 and 1<k<M(n), where M(n) = number of terms in row n = n*(n-1)/2 - (n mod 2); here, the q-factorial of n is denoted [n]_q! = Product_{j=1..n} (1-q^j)/(1-q). - Paul D. Hanna, Jul 07 2008
From Paul D. Hanna, Jun 20 2009: (Start)
For row n>1, the sum over powers of the n-th root of unity = -1:
-1 = Sum_{k=1..n*(n-1)/2} T(n,k)*exp(2*Pi*I*k/n), where I^2=-1.
(End)
EXAMPLE
The table begins:
==============================================================================
k=...|.1.|.2.|.3.|..4.|..5.|..6.|..7.|..8.|..9.|.10.|.11.|.12.|.13.|.14.|.15.|
==============================================================================
n=2..|.1.|
n=3..|.1.|.1.|
n=4..|.1.|.2.|.2.|..2.|..1.|..1.|
n=5..|.1.|.3.|.5.|..7.|..8.|..8.|..6.|..4.|..2.|
n=6..|.1.|.4.|.9.|.16.|.24.|.32.|.37.|.38.|.35.|.28.|.20.|.12.|..6.|..2.|..1.|
===============================================================================
Number of terms in rows 2..22: [1,2,6,9,15,20,28,35,45,54,66,77,91,104,120,135,153,170,190,209,231].
From Paul D. Hanna, Jun 20 2009: (Start)
For row n=4, the sum over powers of I, a 4th root of unity, is:
1*I + 2*I^2 + 2*I^3 + 2*I^4 + 1*I^5 + 1*I^6 = -1. (End)
MATHEMATICA
T[n_, k_] := SeriesCoefficient[QFactorial[n, q] Sum[(-1)^m q^(m(m-1)/2)/ QFactorial[m, q], {m, 0, n}], {q, 0, k}];
Table[T[n, k], {n, 2, 8}, {k, 1, n(n-1)/2 - Mod[n, 2]}] // Flatten (* Jean-François Alcover, Jul 26 2018 *)
PROG
(PARI) T(n, k)=if(k<1 || k>n*(n-1)/2-(n%2), 0, polcoeff( prod(j=1, n, (1-q^j)/(1-q))*sum(k=0, n, (-1)^k*q^(k*(k-1)/2)/if(k==0, 1, prod(j =1, k, (1-q^j)/(1-q)))), k, q)) \\ Paul D. Hanna, Jul 07 2008
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Jonathan Vos Post, Jun 13 2008
EXTENSIONS
More terms from Paul D. Hanna, Jul 07 2008
STATUS
approved