A143947 Triangle read by rows: T(n,k) is the number of permutations of [n] for which the sum of the positions of the right-to-left minima is k (1 <= k <= n*(n+1)/2).

1, 0, 1, 1, 0, 0, 2, 1, 2, 1, 0, 0, 0, 6, 2, 3, 7, 2, 3, 1, 0, 0, 0, 0, 24, 6, 8, 14, 27, 10, 9, 14, 3, 4, 1, 0, 0, 0, 0, 0, 120, 24, 30, 46, 68, 142, 41, 53, 50, 73, 23, 17, 23, 4, 5, 1, 0, 0, 0, 0, 0, 0, 720, 120, 144, 204, 270, 436, 834, 260, 256, 351, 310, 463, 148, 145, 118, 148, 40

Offset: 1

Author

Emeric Deutsch, Sep 22 2008

Keywords

• nonn
• tabf

Comments

Row n contains n(n+1)/2 entries, first n-1 of which are 0. Sum of entries in row n = n! = A000142(n).

Sum of entries in column n = A143948(n).

T(n,n) = (n-1)!.

Sum_{k=n..n(n+1)/2} k*T(n,k) = A001705(n).

Examples

			T(4,6) = 3 because we have 4132, 3142 and 2143 with right-to-left minima at positions 2 and 4.
Triangle starts:
  1;
  0,  1,  1;
  0,  0,  2,  1,  2,  1;
  0,  0,  0,  6,  2,  3,  7,  2,  3,  1;
  0,  0,  0,  0, 24,  6,  8, 14, 27, 10,  9, 14,  3,  4,  1;
  ...
		

Links

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

Crossrefs

Cf. A000142, A000217, A001705, A143946, A143948.

T(n,2n) gives A368678.

Row maxima give A367594.

Programs

Maple

P:=proc(n) options operator, arrow: sort(expand(product(t^(n-j)+j,j=0..n-1))) 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

T[n_] := CoefficientList[Product[n-k+t^k, {k, 1, n-1}] t^(n-1), t];
Array[T, 10] // Flatten (* Jean-François Alcover, Feb 14 2021 *)

Formula

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