0,4

Number of permutations of 1,2,...,k,n+1,n+2,...,2n-k that have no agreements with 1,...,n. For example consider 1234 and 1256, then n=4 and k=2, so T(4,2)=14. Compare A000255 for the case k=1. - Jon Perry (perry(AT)globalnet.co.uk), Jan 23 2004

Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 21 2009: (Start)

T(n-1,k-1) is the number of non-derangements of {1,2,...,n} having smallest fixed point equal to k. Example: T(3,1)=4 because we have 4213, 4231, 3214, and 3241 (the permutations of {1,2,3,4} having smallest fixed equal to 2).

Row sums give the number of non-derangement permutations of {1,2,...,n} (A002467).

Mirror image of A068106.

Closely related to A134830, where each row has an extra term (see the Charalambides reference).

(End)

Sum[(k+1)*T(n,k), k=0..n]=A155521(n+1). - [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 18 2009]

J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122.

E. Deutsch and S. Elizalde, The largest and the smallest fixed points of permutations, arXiv:0904.2792v1, 2009. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 21 2009]

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 176, Table 5.3. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 21 2009]

Index entries for sequences related to factorial numbers

t(n, k) =t(n, k-1)-t(n-1, k-1) =t(n, k+1)-t(n-1, k) =n*t(n-1, k)+k*t(n-2, k-1) =(n-1)*t(n-1, k-1)+(k-1)*t(n-2, k-2) =A060475(n, k)*(n-k)! - Henry Bottomley (se16(AT)btinternet.com), Mar 16 2001

T(n, k) = Sum_{ j>= 0} (-1)^j * binomial(k, j)*(n-j)! . - Philippe DELEHAM, May 29 2005

T(n,k)=Sum[d(n-j)*binom(n-k,j), j=0..n-k], where d(i)=A000166(i) are the derangement numbers. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 17 2009

1; 1,0; 2,1,1; 6,4,3,2; 24,18,14,11,9; 120,96,78,64,53,44; ...

d[0] := 1: for n to 15 do d[n] := n*d[n-1]+(-1)^n end do: T := proc (n, k) if k <= n then sum(binomial(n-k, j)*d[n-j], j = 0 .. n-k) else 0 end if end proc: for n from 0 to 9 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 17 2009]

Columns give A000142, A001563, A001564, etc. Cf. A047922.

See A068106 for another version of this triangle.

Orthogonal columns: A000166, A000255, A055790. Main diagonal A033815.

A068106, A002467, A134830 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 21 2009]

Cf. A155521.

Sequence in context: A158389 A103880 A135899 this_sequence A144655 A140956 A166919

Adjacent sequences: A047917 A047918 A047919 this_sequence A047921 A047922 A047923

nonn,tabl,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).