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Search: id:A143946
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| A143946 |
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Triangle read by rows: T(n,k) is the number of permutations of [n] for which the sum of the positions of the left-to-right maxima is k (1<=k<=n(n+1)/2). |
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+0
2
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| 1, 1, 0, 1, 2, 0, 2, 1, 0, 1, 6, 0, 6, 3, 2, 3, 2, 1, 0, 1, 24, 0, 24, 12, 8, 18, 8, 10, 3, 6, 3, 2, 1, 0, 1, 120, 0, 120, 60, 40, 90, 64, 50, 39, 42, 23, 28, 13, 10, 8, 6, 3, 2, 1, 0, 1, 720, 0, 720, 360, 240, 540, 384, 420, 234, 372, 198, 208, 168, 124, 98, 75, 60, 35, 34, 13, 16, 8, 6, 3
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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Row n contains n(n+1)/2 entries.
Sum of entries in row n = n!=A000142(n).
T(n,1)=T(n,3)=(n-1)! for n>=2.
Sum(k*T(n,k),k=1..n(n+1)/2)=n!*n = A001563(n).
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REFERENCES
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I. Kortchemski, Asymptotic behavior of permutation records, arXiv: 0804.0446v2 [math.CO], 18 May 2008.
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FORMULA
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Generating polynomial of row n is t(t^2+1)(t^3+2)...(t^n+n-1).
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EXAMPLE
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T(4,6)=3 because we have 1243, 1342 and 2341 with left-to-right maxima at positions 1,2,3.
Triangle starts:
1;
1,0,1;
2,0,2,1,0,1;
6,0,6,3,2,3,2,1,0,1;
24,0,24,12,8,18,8,10,3,6,3,2,1,0,1
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MAPLE
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P:=proc(n) options operator, arrow: sort(expand(product(t^j+j-1, j=1..n))) end proc: for n to 7 do seq(coeff(P(n), t, i), i=1..(1/2)*n*(n+1)) end do; # yields sequence in triangular form
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CROSSREFS
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A000142, A001563
Sequence in context: A101674 A100820 A038760 this_sequence A035394 A067167 A096810
Adjacent sequences: A143943 A143944 A143945 this_sequence A143947 A143948 A143949
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 21 2008
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