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%I #24 Oct 24 2014 04:19:50
%S 2520,30240,226800,1367520,7271880,35692800,165957792,742822080,
%T 3234711480,13803744864,58021888080,241116750624,993313349544,
%U 4064913201216,16549636147968,67112688842496,271323921459096,1094303232174240,4405390451382960,17709538489849440
%N Number of doubly-surjective functions f:[n]->[4].
%C Fourth column of A200091: A200091(n,4)=a(n).
%C Also, a(n) is (i) the number of length-n words on the alphabet A, B, C, and D with each letter of the alphabet occurring at least twice; (ii) the number of ways to distribute n different toys to 4 children so that each child gets at least two toys; (iii) the number of ways to put n numbered balls into 4 labeled boxes so that each box gets at least two balls; (iv) the number of n-digit positive integers consisting of only digits 1,2,3, and 4 with each of these digits appearing at least twice.
%C A doubly-surjective function f:D->C is such that the pre-image set f^-1(y) has size at least 2 for each y in C.
%C Triangle A200091 provides the number of doubly-surjective functions f from a set of size n onto a set of size k. Hence a(n) is column 4 of A200091.
%H Dennis Walsh, <a href="http://capone.mtsu.edu/dwalsh/DOUBSURJ.pdf">Notes on doubly-surjective finite functions</a>
%F a(n) = 4^n-4*3^n-4*n*3^(n-1)+(9*n+3*n^2)*2^(n-1)+6*2^n-4-8*n-4*n^3;
%F a(n) = sum(n!/(i!j!k!m!) over <i,j,k,m> such that i,j,k, and m are all at least 2 and i+j+k+m=n.
%F E.g.f.: (exp(x)-x-1)^4.
%F a(n) = 24*A058844(n). - _Alois P. Heinz_, Apr 10 2013
%F G.f.: 24*x^8*(288*x^6-1560*x^5+3500*x^4-4130*x^3+2625*x^2-840*x+105) / ((x-1)^4*(2*x-1)^3*(3*x-1)^2*(4*x-1)). - _Colin Barker_, Jun 04 2013
%e a(9) = 30240 since there are 30240 ways to distribute 9 different toys to 4 children so that each child gets at least 2 toys. One child must get 3 toys and the other children get 2 toys each. There are 4 ways to pick the lucky kid. There are C(9,3) ways to choose the 3 toys for the lucky kid. There are 6!/(2!)^3 ways to distribute the remaining 6 toys among the 3 kids. We obtain 4*C(9,3)*6!/8=30240.
%p seq(eval(diff((exp(x)-x-1)^4,x$n),x=0),n=8..40);
%t nn=27; Drop[Range[0,nn]! CoefficientList[Series[(Exp[x]-x-1)^4, {x,0,nn}], x], 8] (* _Geoffrey Critzer_, Sep 28 2013 *)
%o (PARI) x='x+O('x^66); Vec(serlaplace((exp(x)-x-1)^4)) /* _Joerg Arndt_, Apr 10 2013 */
%Y Cf. A224541, A052515.
%K nonn,easy
%O 8,1
%A _Dennis P. Walsh_, Apr 09 2013
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