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
Examples
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)
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.
Programs
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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 *) -
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
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)
Extensions
More terms from Paul D. Hanna, Jul 07 2008
Comments