A086363 - OEIS

A086363

Array T(m,n) read by antidiagonals: if X and Y are two (possibly empty) finite sets with m and n elements respectively and Z is the disjoint union of X and Y, then T(m,n) is the number of self-inverse partial functions f:Z ->Z which do not fix any element of Y.

1, 1, 2, 2, 3, 5, 4, 6, 9, 14, 10, 14, 20, 29, 43, 26, 36, 50, 70, 99, 142, 76, 102, 138, 188, 258, 357, 499, 232, 308, 410, 548, 736, 994, 1351, 1850, 764, 996, 1304, 1714, 2262, 2998, 3992, 5343, 7193

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..44.

FORMULA

T(m, n) = T(m, n-1) + m*T(m-1, n-1) + (n-1)*T(m, n-2) for m>0, n>1; T(m, 0) = b(m); T(m, 1) = b(m) + m*b(m-1); T(0, n) = c(n); where sequences b and c are A005425 and A000085 respectively.

EXAMPLE

T(1,2)=6: If we let X={1}, Y={2,3}, so Z={1,2,3} and the relevant partial functions f:Z ->Z which do not fix either 2 or 3 are (-,-,-), (1,-,-), (-,3,2), (1,3,2), (2,1,-), (3,-,1). Here a partial function f:Z ->Z is displayed as (f(1),f(2),f(3)).

Array begins:

1, 1, 2, 4, 10, 26, 76, 232, 764, ...

2, 3, 6, 14, 36, 102, 308, 996, 3384, ...

5, 9, 20, 50, 138, 410, 1304, 4380, 15500, ...

14, 29, 70, 188, 548, 1714, 5684, 19880, 72808, ...

PROG

(PARI) T(m, n)={ if(m, if(n>1, T(m, n-1)+m*T(m-1, n-1)+(n-1)*T(m, n-2), A005425(m)+if(n, A005425(m-1)*m)), A000085(n))} \\ M. F. Hasler, Jan 13 2012

for(i=1, 9, for(j=1, i, print1(T(j-1, i-j)", "))) /* list values by antidiagonals */

CROSSREFS

Sequence in context: A036716 A026399 A117267 * A174094 A360461 A284114

Adjacent sequences: A086360 A086361 A086362 * A086364 A086365 A086366

KEYWORD

easy,nonn,tabl

AUTHOR

James East, Sep 04 2003

EXTENSIONS

Corrected and extended by Philippe Deléham, Dec 31 2011

Values double-checked using the given PARI/GP code by M. F. Hasler, Jan 13 2012

STATUS

approved