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A019575
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Place n distinguishable balls in n boxes (in n^n ways); let T(n,k) = number of ways that the maximum in any box is k, for 1 <= k <= n; sequence gives triangle of numbers T(n,k).
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8
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1, 2, 2, 6, 18, 3, 24, 180, 48, 4, 120, 2100, 800, 100, 5, 720, 28800, 14700, 2250, 180, 6, 5040, 458640, 301350, 52920, 5292, 294, 7, 40320, 8361360, 6867840, 1342600, 153664, 10976, 448, 8, 362880, 172141200, 172872000, 36991080, 4644864, 387072, 20736, 648, 9
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OFFSET
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1,2
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COMMENTS
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T(n,k) is the number of endofunctions on [n] such that the maximal cardinality of the nonempty preimages equals k. - Alois P. Heinz, Jul 31 2014
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LINKS
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FORMULA
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Let f(n,k,b) = number of ways to place b balls to n boxes, where the max in any box is not larger than k. Then T(n,k) = f(n,k,n) - f(n,k-1,n). We have:
f(n, k, b)=local(i); if(n==0, return(b==0));return(sum(i=0, min(k, b), binomial(b, i)*f(n-1, k, b-i))).
T(n,k) = f(n,k,n) - f(n,k-1,n).
(end)
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EXAMPLE
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Triangle begins:
1;
2, 2;
6, 18, 3;
24, 180, 48, 4;
120, 2100, 800, 100, 5;
720, 28800, 14700, 2250, 180, 6;
5040, 458640, 301350, 52920, 5292, 294, 7;
40320, 8361360, 6867840, 1342600, 153664, 10976, 448, 8;
362880, 172141200, 172872000, 36991080, 4644864, 387072, 20736, 648, 9;
...
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-j, i-1, k)/j!, j=0..min(k, n))))
end:
T:= (n, k)-> n!* (b(n$2, k) -b(n$2, k-1)):
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MATHEMATICA
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f[0, _, b_] := Boole[b == 0]; f[n_, k_, b_] := f[n, k, b] = Sum[ Binomial[b, i]*f[n - 1, k, b - i], {i, 0, Min[k, b]}]; t[n_, k_] := f[n, k, n] - f[n, k - 1, n]; Flatten[ Table[ t[n, k], {n, 1, 9}, {k, 1, n}]] (* Jean-François Alcover, Mar 09 2012, after Robert Gerbicz *)
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PROG
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(PARI)
/*setup memoization table for args <= M. Could be done dynamically inside f() */
M=10; F=vector(M, i, vector(M, i, vector(M)));
f(n, k, b)={ (!n|!b|!k) & return(!b); F[n][k][b] & return(F[n][k][b]);
F[n][k][b]=sum(i=0, min(k, b), binomial(b, i)*f(n-1, k, b-i)) }
T(n, k)=f(n, k, n)-f(n, k-1, n)
for(n=1, 9, print(vector(n, k, T(n, k))))
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CROSSREFS
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Cf. A019576. See A180281 for the case when the balls are indistinguishable.
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KEYWORD
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AUTHOR
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Lee Corbin (lcorbin(AT)tsoft.com)
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EXTENSIONS
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STATUS
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approved
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