A144091 T(n,k) is the number of partial bijections (or subpermutations) of an n-element set of height k (height(alpha) = |Im(alpha)|) and with exactly 2 fixed points

1, 3, 0, 6, 12, 6, 10, 60, 90, 20, 15, 180, 630, 660, 135, 21, 420, 2730, 6720, 5565, 924, 28, 840, 8820, 39760, 76020, 51912, 7420, 36, 1512, 23436, 168840, 585900, 917784, 533988, 66744

Offset

2,2

Links

Table of n, a(n) for n=2..37.

A. Laradji and A. Umar, Combinatorial results for the symmetric inverse semigroup, Semigroup Forum 75, (2007), 221-236.

Formula

T(n,k) = (n!/2!(n-k)!)sum(m=0,k-2,(-1^m/m!)C(n-2-m,k-2-m)).

Example

T(4,2) = 6 because there are exactly 6 partial bijections (on a 4-element set) with exactly 2 fixed points and of height 2, namely: the 6 partial identities on 2-element subsets of the 4-element set.

Prog

(PARI) T(n, k) = (n!/2!*(n-k)!)*sum(m=0, k-2, ((-1)^m/m!)*binomial(n-2-m, k-2-m))
for (n=2, 10, for (k=2, n, print1(T(n, k), ", "))) \\ Michel Marcus, Apr 27 2016

Crossrefs

Row sums are A144087.

Sequence in context: A141434 A077911 A057381 * A019145 A059684 A270509

Adjacent sequences: A144088 A144089 A144090 * A144092 A144093 A144094

Keyword

nonn,tabl

Author

Abdullahi Umar, Sep 11 2008

Status

approved