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%I #22 Aug 22 2021 14:56:35
%S 1,1,2,2,3,5,4,6,9,14,10,14,20,29,43,26,36,50,70,99,142,76,102,138,
%T 188,258,357,499,232,308,410,548,736,994,1351,1850,764,996,1304,1714,
%U 2262,2998,3992,5343,7193
%N 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.
%F 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.
%e 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)).
%e Array begins:
%e 1, 1, 2, 4, 10, 26, 76, 232, 764, ...
%e 2, 3, 6, 14, 36, 102, 308, 996, 3384, ...
%e 5, 9, 20, 50, 138, 410, 1304, 4380, 15500, ...
%e 14, 29, 70, 188, 548, 1714, 5684, 19880, 72808, ...
%o (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
%o for(i=1,9,for(j=1,i,print1(T(j-1,i-j)","))) /* list values by antidiagonals */
%K easy,nonn,tabl
%O 0,3
%A _James East_, Sep 04 2003
%E Corrected and extended by _Philippe Deléham_, Dec 31 2011
%E Values double-checked using the given PARI/GP code by _M. F. Hasler_, Jan 13 2012
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