A261765 Triangle read by rows: T(n,k) is the number of subpermutations of an n-set, whose orbits are each of size at most k with at least one orbit of size exactly k, and without fixed points. Equivalently, T(n,k) is the number of partial derangements of an n-set each of whose orbits is of size at most k with at least one orbit of size exactly k, and without fixed points.

1, 1, 0, 1, 0, 3, 1, 0, 9, 8, 1, 0, 45, 32, 30, 1, 0, 165, 320, 150, 144, 1, 0, 855, 2240, 1800, 864, 840, 1, 0, 3843, 17360, 18900, 12096, 5880, 5760, 1, 0, 21819, 146048, 195300, 145152, 94080, 46080, 45360, 1, 0, 114075, 1256192, 2120580, 1959552, 1270080, 829440, 408240, 403200

Offset

0,6

Comments

T(n,n) is A261766. Sum of rows is A144085.

References

A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460.

Links

Table of n, a(n) for n=0..54.

Formula

T(n,k) = A261762(n,k) - A261762(n,k-1).

Example

T(n,1) = 0 because there is no (partial) derangement with an orbit of size 1.
T(3,2) = 9 because there are 9 subpermutations on {1,2,3}, whose orbits are each of size at most 2 with at least one orbit of size exactly 2, and without fixed points, namely: (1 2 --> 2 1), (1 3 --> 3 1), (2 3 --> 3 2), (1-->2), (1-->3), (2-->1), (2-->3), (3-->1), (3-->2).
Triangle starts:
1;
1, 0;
1, 0, 3;
1, 0, 9, 8;
1, 0, 45, 32, 30;
1, 0, 165, 320, 150, 144;
1, 0, 855, 2240, 1800, 864, 840;
...

Crossrefs

Cf. A157400, A261762, A261763, A261764, A261766, A261767.

Sequence in context: A094753 A355350 A221713 * A079669 A143398 A202995

Adjacent sequences: A261762 A261763 A261764 * A261766 A261767 A261768

Keyword

nonn,tabl

Author

Samira Stitou, Sep 21 2015

Extensions

More terms from Alois P. Heinz, Nov 04 2015

Status

approved