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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
OFFSET
0,3
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
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