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A217877
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Triangle read by rows: minimum inversion terminator in rooted labeled trees.
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1
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1, 2, 1, 6, 8, 2, 24, 75, 20, 6, 120, 864, 216, 72, 24, 720, 12005, 2744, 882, 336, 120, 5040, 196608, 40960, 12288, 4608, 1920, 720, 40320, 3720087, 708588, 196830, 69984, 29160, 12960, 5040, 362880, 80000000, 14000000, 3600000, 1200000, 480000, 216000, 100800, 40320
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OFFSET
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2,2
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COMMENTS
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T(n,k) is the number of trees on vertex set [0,n-1], rooted at 0, with minimum inversion terminator = k if k>=1, with no inversion terminators if k=0. An inversion is a pair i,j of vertices with j a descendant of i and i>j; j is then an inversion terminator.
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LINKS
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FORMULA
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T(n,0) = (n-1)!, T(n,k) = k!*(n-k-1)*n^(n-k-2) for 1<=k<=n-2.
Proof. For any given increasing tree T on [0,k], the number of rooted-at-0 trees on [0,n-1] that contain T is (k+1)n^(n-k-2) [J. W. Moon, Counting Labelled Trees (1970), Sec. 6.2]. Hence, since there are k! increasing trees on [0,k] [R. H. Stanley, Enumerative Combinatorics, Vol. 1, (1986), Sec. 1.3.16], the number of trees on [0,n-1] that contain *some* increasing tree on [0,k] is (k+1)!n^(n-k-2). But the minimum inversion terminator is k precisely when the tree contains some increasing tree on [0,k-1] but none on [0,k]. The number of such trees is therefore k!n^(n-k-1) - (k+1)!n^(n-k-2) = T(n,k) (for k>=1). QED.
This gives a nice combinatorial interpretation of the identity n^(n-2) = (n-1)! + Sum_{k=1..n-2} k!(n-k-1)n^(n-k-2). The identity is easy to establish analytically, of course, because the sum is telescoping.
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EXAMPLE
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Triangle starts at row n=2:
1;
2, 1;
6, 8, 2;
24, 75, 20, 6;
120, 864, 216, 72, 24;
720, 12005, 2744, 882, 336, 120;
5040, 196608, 40960, 12288, 4608, 1920, 720;
...
T(4,2)=2 counts 0->3->2, 0->1 and 0->1->3->2, in both of which the minimum (and only) inversion terminator is 2.
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MATHEMATICA
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Table[If[k==0, (n-1)!, k!(n-k-1)n^(n-k-2)], {n, 2, 12}, {k, 0, n-2}]
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PROG
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(PARI) T(n, k) = {if(!k, (n-1)!, k!*(n-k-1)*n^(n-k-2))}
{ for(n=2, 10, for(k=0, n-2, print1(T(n, k), ", ")); print) } \\ Andrew Howroyd, Apr 28 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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