A342645 Triangle read by rows: T(n,k) gives n! times the coefficient of x^k in the polynomial that describes the number of permutations on x letters with major index n.

1, -1, 1, -2, -1, 1, 0, -7, 0, 1, 0, -14, -13, 2, 1, 120, -46, -65, -15, 5, 1, 0, 516, -356, -165, -5, 9, 1, 5040, 1392, 266, -1421, -280, 28, 14, 1, 0, 46320, 3772, -5740, -3871, -280, 98, 20, 1, 0, 215280, 212724, -26272, -31437, -7791, 126, 222, 27, 1

Offset: 0

Author

Peter Kagey, Mar 17 2021

Keywords

• sign
• tabl

Comments

This n-th row describes a polynomial that eventually agrees with the n-th column of A008302.

Conjecture: For each m, T(n,n-m) is a polynomial of degree 2m whose leading coefficient is abs(A290030(m)/A053657(m+1)).

Examples

			n\k |    0       1       2       3       4      5    6    7   8  9
----+--------------------------------------------------------------
  0 |    1;
  1 |   -1,      1;
  2 |   -2,     -1,      1;
  3 |    0,     -7,      0,      1;
  4 |    0,    -14,    -13,      2,      1;
  5 |  120,    -46,    -65,    -15,      5,     1;
  6 |    0,    516,   -356,   -165,     -5,     9,   1;
  7 | 5040,   1392,    266,  -1421,   -280,    28,  14,   1;
  8 |    0,  46320,   3772,  -5740,  -3871,  -280,  98,  20,  1;
  9 |    0, 215280, 212724, -26272, -31437, -7791, 126, 222, 27, 1;
For n = 4, the polynomial that describes the 4th column of A008302 is
A008302(x,4) = (-14x -13x^2 +2x^3 + x^4)/4! = Sum_{j=0..4} (T(j,4)*x^j)/4!.
		

Links

• Peter Kagey, Rows n = 0..100, flattened
• Mike Earnest, Does the number of permutations in S_n with major index equal to k, satisfy a degree k polynomial?, Mathematics Stack Exchange answer.

Crossrefs

Cf. A008302, A053657, A290030.

Programs

Mathematica

A008302T[0, 0] := 1; A008302T[-1, k_] := 0;
A008302T[n_, k_] := (A008302T[n, k] = If[0 <= k <= n*(n - 1)/2, A008302T[n, k - 1] + A008302T[n - 1, k] - A008302T[n - 1, k - n], 0]);
A342645Row[n_] := (A342645Row[n] = Expand[n!*InterpolatingPolynomial[Table[{m, A008302T[m, n]}, {m, n, 2*n + 2}], x]]);
A342645T[n_, k_] := Coefficient[A342645Row[n], x, k];

Formula

Conjectures:

T(n,n) = 1.

T(n,n-1) = (-3n + n^2)/2.

T(n,n-2) = (-2n + 21n^2 - 22n^3 + 3n^4)/24.

T(n,n-3) = (96n - 134n^2 + 13n^3 + 37n^4 - 13n^5 + n^6)/48.