1,5

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 (njas(AT)research.att.com), 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). [From Emeric Deutsch (deutsch(AT)duke.poly.edu), May 04 2009]

Paul D. Hanna, Table of n, A139755(m,k), as a flattened table for rows m = 2..22

William Y. C. Chen and David G. L. Wang, The Limiting Distributions of the Coefficients of the q-Derangement Number

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 (pauldhanna(AT)juno.com), Jul 07 2008

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), 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)

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].

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Jun 20 2009: (Start)

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

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

(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 (pauldhanna(AT)juno.com), Jul 07 2008

Cf. A000166.

Cf. diagonals: A141753, A141754.

Cf. A152290. [From Paul D. Hanna (pauldhanna(AT)juno.com), Jun 20 2009]

Sequence in context: A144110 A076490 A124278 this_sequence A104637 A058745 A108393

Adjacent sequences: A139752 A139753 A139754 this_sequence A139756 A139757 A139758

nonn,tabl

Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 13 2008

More terms from Paul D. Hanna (pauldhanna(AT)juno.com), Jul 07 2008