OFFSET
2,2
COMMENTS
T(n,k) = number of permutations of n elements with k fixed points.
T(n,n-1)=0 and T(n,n)=1 are omitted from the array. - Geoffrey Critzer, Nov 28 2011.
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 194.
Kaufmann, Arnold. "Introduction a la combinatorique en vue des applications." Dunod, Paris, 1968. See p. 92.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.
LINKS
T. D. Noe, Rows n=2..50, flattened
FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation
I. Kaplansky, Symbolic solution of certain problems in permutations, Bull. Amer. Math. Soc., 50 (1944), 906-914.
FORMULA
E.g.f. for column k: (x^k/k!)(exp(-x)/(1-x)). - Geoffrey Critzer, Nov 28 2011
Row generating polynomials appear to be given by -1 + sum {k = 0..n} (-1)^(n+k)*C(n,k)*(1+k*x)^(n-k)*(2+(k-1)*x)^k. - Peter Bala, Dec 29 2011
EXAMPLE
Triangle begins:
1
2 3
9 8 6
44 45 20 10
265 264 135 40 15
1854 1855 924 315 70 21
14833 14832 7420 2464 630 112 28
133496 133497 66744 22260 5544 1134 168 36
...
MAPLE
T:= proc(n, k) T(n, k):= `if`(k=0, `if`(n<2, 1-n, (n-1)*
(T(n-1, 0)+T(n-2, 0))), binomial(n, k)*T(n-k, 0))
end:
seq(seq(T(n, k), k=0..n-2), n=2..12); # Alois P. Heinz, Mar 17 2013
MATHEMATICA
Prepend[Flatten[f[list_]:=Select[list, #>1&]; Map[f, Drop[Transpose[Table[d = Exp[-x]/(1 - x); Range[0, 10]! CoefficientList[Series[d x^k/k!, {x, 0, 10}], x], {k, 0, 8}]], 3]]], 1] (* Geoffrey Critzer, Nov 28 2011 *)
PROG
(PARI) T(n, k)= if(k<0 || k>n, 0, n!/k!*sum(i=0, n-k, (-1)^i/i!))
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
Comments and more terms from Michael Somos, Apr 26 2000
STATUS
approved