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A144088 T(n,k) is the number of partial bijections (or subpermutations) of an n-element set with exactly k fixed points. 3
1, 1, 1, 4, 2, 1, 18, 12, 3, 1, 108, 72, 24, 4, 1, 780, 540, 180, 40, 5, 1, 6600, 4680, 1620, 360, 60, 6, 1, 63840, 46200, 16380, 3780, 630, 84, 7, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..35.

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

A. Umar, Some combinatorial problems in the theory of symmetric ..., Algebra Disc. Math. 9 (2010) 115-126

FORMULA

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

(n-k)*T(n,k) = n*(2n-2k-1)*T(n-1,k) - n*(n-1)*(n-k-3)*T(n-2,k) - n*(n-1)*(n-2)*T(n-3,k), T(k,k)=1 and T(n,k)=0 if n < k.

EXAMPLE

T(3,1) = 12 because there are exactly 12 partial bijections (on a 3-element set) with exactly 1 fixed point, namely: (1)->(1), (2)->(2), (3)->(3), (1,2)->(1,3), (1,2)->(3,2), (1,3)->(1,2), (1,3)->(2,3), (2,3)->(2,1), (2,3)->(1,3), (1,2,3)->(1,3,2), (1,2,3)->(3,2,1), (1,2,3)->(2,1,3) - the mappings are coordinate-wise.

MATHEMATICA

max = 7; f[x_, k_] := (x^k/k!)*(Exp[x^2/(1-x)]/(1-x)); t[n_, k_] := n!*SeriesCoefficient[ Series[ f[x, k], {x, 0, max}], n]; Flatten[ Table[ t[n, k], {n, 0, max}, {k, 0, n}]](* Jean-Fran├žois Alcover, Mar 12 2012, from e.g.f. by Joerg Arndt *)

CROSSREFS

T(n, 0) = A144085, T(n, 1) = A144086, T(n, 2) = A144087

Sum of rows is A002720.

Sequence in context: A264535 A256039 A152391 * A039948 A111536 A111559

Adjacent sequences:  A144085 A144086 A144087 * A144089 A144090 A144091

KEYWORD

nice,nonn,tabl

AUTHOR

Abdullahi Umar, Sep 11 2008, Sep 16 2008

STATUS

approved

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Last modified April 18 10:35 EDT 2019. Contains 322209 sequences. (Running on oeis4.)