A139755 Table of q-derangement numbers of type A, by rows.

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, 6, 2, 1, 1, 5, 14, 30, 54, 86, 123, 160, 191, 210, 214, 202, 176, 141, 104, 69, 41, 21, 9, 3, 1, 6, 20, 50, 104, 190, 313, 473, 663, 868, 1068, 1240, 1362, 1417, 1398, 1307, 1157, 968

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

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, 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

Cf. A000166, A152290.

Cf. diagonals: A141753, A141754.

Sequence in context: A302111 A124278 A253187 * A213366 A212648 A255507

Adjacent sequences: A139752 A139753 A139754 * A139756 A139757 A139758

Keyword

nonn,tabf

Author

Jonathan Vos Post, Jun 13 2008

Extensions

More terms from Paul D. Hanna, Jul 07 2008

Status

approved