|
|
A202363
|
|
Triangular array read by rows: T(n,k) is the number of inversion pairs ( p(i) < p(j) with i>j ) that are separated by exactly k elements in all n-permutations (where the permutation is represented in one line notation); n>=2, 0<=k<=n-2.
|
|
1
|
|
|
1, 6, 3, 36, 24, 12, 240, 180, 120, 60, 1800, 1440, 1080, 720, 360, 15120, 12600, 10080, 7560, 5040, 2520, 141120, 120960, 100800, 80640, 60480, 40320, 20160, 1451520, 1270080, 1088640, 907200, 725760, 544320, 362880, 181440, 16329600, 14515200, 12700800, 10886400, 9072000, 7257600, 5443200, 3628800, 1814400
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,2
|
|
COMMENTS
|
|
|
LINKS
|
|
|
FORMULA
|
E.g.f.: x^2/2 * (1/(1-x)^2)* (1/(1-y*x)).
|
|
EXAMPLE
|
T(3,1) = 3 because from the permutations (given in one line notation): (2,3,1), (3,1,2), (3,2,1) we have respectively 3 inversion pairs (1,2), (2,3) and (1,3) which are all separated by 1 element.
Triangle T(n,k) begins:
1;
6, 3;
36, 24, 12;
240, 180, 120, 60;
1800, 1440, 1080, 720, 360;
15120, 12600, 10080, 7560, 5040, 2520;
141120, 120960, 100800, 80640, 60480, 40320, 20160;
...
|
|
MATHEMATICA
|
nn=10; Range[0, nn]!CoefficientList[Series[x^2/2/(1-x)^2/(1-y x), {x, 0, nn}], {x, y}]//Grid
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|