A143946 - OEIS

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, 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

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OFFSET

1,5

COMMENTS

Row n contains n*(n+1)/2 = A000217(n) entries.

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

LINKS

Alois P. Heinz, Rows n = 1..50, flattened

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).

Sum_{k=1..n*(n+1)/2} (n*(n+1)/2-k) * T(n,k) = A001804(n). - Alois P. Heinz, Dec 19 2023

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, 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

# second Maple program:

b:= proc(n) option remember; `if`(n=0, 1,

expand(b(n-1)*(x^n+n-1)))

end:

T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):

seq(T(n), n=1..7); # Alois P. Heinz, Aug 05 2020

MATHEMATICA

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

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

CROSSREFS

Cf. A000142, A000217, A001563, A001804, A143947.

T(n,n) gives A368246.