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A139755 Table of q-derangement numbers of type A, by rows. 4
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified August 9 22:52 EDT 2020. Contains 336335 sequences. (Running on oeis4.)