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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. 0
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

E.g. 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 A284114 A323480

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

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Last modified February 18 15:04 EST 2020. Contains 332019 sequences. (Running on oeis4.)