A139755 - OEIS

%I #23 Jul 26 2018 10:08:55

%S 1,1,1,1,2,2,2,1,1,1,3,5,7,8,8,6,4,2,1,4,9,16,24,32,37,38,35,28,20,12,

%T 6,2,1,1,5,14,30,54,86,123,160,191,210,214,202,176,141,104,69,41,21,9,

%U 3,1,6,20,50,104,190,313,473,663,868,1068,1240,1362,1417,1398,1307,1157,968

%N Table of q-derangement numbers of type A, by rows.

%C 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.

%C Number of terms in row n appears to be A084265(n+2). - _N. J. A. Sloane_, Jul 20 2008

%C 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

%H Paul D. Hanna, <a href="/A139755/b139755.txt">Table of n, A139755(m,k), as a flattened table for rows m = 2..22</a>

%H William Y. C. Chen and David G. L. Wang, <a href="http://arxiv.org/abs/0806.2092">The Limiting Distributions of the Coefficients of the q-Derangement Number</a>, arXiv:0806.2092 [math.CO], 2008.

%F 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

%F From _Paul D. Hanna_, Jun 20 2009: (Start)

%F For row n>1, the sum over powers of the n-th root of unity = -1:

%F -1 = Sum_{k=1..n*(n-1)/2} T(n,k)*exp(2*Pi*I*k/n), where I^2=-1.

%F (End)

%e The table begins:

%e ==============================================================================

%e k=...|.1.|.2.|.3.|..4.|..5.|..6.|..7.|..8.|..9.|.10.|.11.|.12.|.13.|.14.|.15.|

%e ==============================================================================

%e n=2..|.1.|

%e n=3..|.1.|.1.|

%e n=4..|.1.|.2.|.2.|..2.|..1.|..1.|

%e n=5..|.1.|.3.|.5.|..7.|..8.|..8.|..6.|..4.|..2.|

%e n=6..|.1.|.4.|.9.|.16.|.24.|.32.|.37.|.38.|.35.|.28.|.20.|.12.|..6.|..2.|..1.|

%e ===============================================================================

%e 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].

%e From _Paul D. Hanna_, Jun 20 2009: (Start)

%e For row n=4, the sum over powers of I, a 4th root of unity, is:

%e 1*I + 2*I^2 + 2*I^3 + 2*I^4 + 1*I^5 + 1*I^6 = -1. (End)

%t T[n_, k_] := SeriesCoefficient[QFactorial[n, q] Sum[(-1)^m q^(m(m-1)/2)/ QFactorial[m, q], {m, 0, n}], {q, 0, k}];

%t Table[T[n, k], {n, 2, 8}, {k, 1, n(n-1)/2 - Mod[n, 2]}] // Flatten (* _Jean-François Alcover_, Jul 26 2018 *)

%o (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

%Y Cf. A000166, A152290.

%Y Cf. diagonals: A141753, A141754.

%K nonn,tabf

%O 1,5

%A _Jonathan Vos Post_, Jun 13 2008

%E More terms from _Paul D. Hanna_, Jul 07 2008