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A143946 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). 1
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; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row n contains n(n+1)/2 entries.

Sum of entries in row n = n! = A000142(n).

LINKS

Table of n, a(n) for n=1..80.

I. Kortchemski, Asymptotic behavior of permutation records, arXiv: 0804.0446 [math.CO], 2008-2009.

FORMULA

T(n,1) = T(n,3) = (n-1)! for n>=2.

Sum_{k=1..n(n+1)/2} k * T(n,k) = n! * n = A001563(n).

Generating polynomial of row n is t(t^2+1)(t^3+2)...(t^n+n-1).

EXAMPLE

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;

MAPLE

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

MATHEMATICA

row[n_] := CoefficientList[Product[t^k + k - 1, {k, 1, n}], t] // Rest;

Array[row, 7] // Flatten (* Jean-Fran├žois Alcover, Nov 28 2017 *)

CROSSREFS

Cf. A000142, A001563.

Sequence in context: A100820 A038760 A245825 * A226860 A244526 A035394

Adjacent sequences:  A143943 A143944 A143945 * A143947 A143948 A143949

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Sep 21 2008

STATUS

approved

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Last modified May 24 19:14 EDT 2020. Contains 334580 sequences. (Running on oeis4.)