A144090 Triangle read by rows: 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 1 fixed point.

1, 2, 0, 3, 6, 3, 4, 24, 36, 8, 5, 60, 210, 220, 45, 6, 120, 780, 1920, 1590, 264, 7, 210, 2205, 9940, 19005, 12978, 1855, 8, 336, 5208, 37520, 130200, 203952, 118664, 14832, 9, 504, 10836, 114408, 630630, 1783656, 2369556, 1201464, 133497

Offset: 1

Author

Abdullahi Umar, Sep 11 2008

Keywords

• nonn
• tabl

Examples

			T(3,2) = 6 because there are exactly 6 partial bijections (on a 3-element set) with exactly 1 fixed point and of height 2, namely: (1,2)->(1,3), (1,2)->(3,2), (1,3)->(1,2), (1,3)->(2,3), (2,3)->(2,1), (2,3)->(1,3)- the mappings are coordinate-wise.
First six rows:
1
2      0
3      6      3
4     24     36       8
5     60    210     220      45
6    120    780    1920    1590    264
		

Links

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

Crossrefs

Rows sums are A144086.

Main diagonal gives A000240.

Programs

Mathematica

Table[(n!/(n - k)!) Sum[ ((-1)^m/m!) Binomial[n - 1 - m, k - 1 - m], {m, 0, k - 1}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Apr 27 2016 *)

PARI

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

Formula

T(n,k) = (n!/(n-k)!)*Sum_{m=0..k-1} ((-1)^m/m!)*C(n-1-m,k-1-m).