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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.

%I A342645 #25 Aug 12 2022 19:10:03
%S A342645 1,-1,1,-2,-1,1,0,-7,0,1,0,-14,-13,2,1,120,-46,-65,-15,5,1,0,516,-356,
%T A342645 -165,-5,9,1,5040,1392,266,-1421,-280,28,14,1,0,46320,3772,-5740,
%U A342645 -3871,-280,98,20,1,0,215280,212724,-26272,-31437,-7791,126,222,27,1
%N 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.
%C A342645 This n-th row describes a polynomial that eventually agrees with the n-th column of A008302.
%C A342645 Conjecture: For each m, T(n,n-m) is a polynomial of degree 2m whose leading coefficient is abs(A290030(m)/A053657(m+1)).
%H A342645 Peter Kagey, <a href="/A342645/b342645.txt">Rows n = 0..100, flattened</a>
%H A342645 Mike Earnest, <a href="https://math.stackexchange.com/a/4065913/121988">Does the number of permutations in S_n with major index equal to k, satisfy a degree k polynomial?</a>, Mathematics Stack Exchange answer.
%F A342645 Conjectures:
%F A342645 T(n,n)   = 1.
%F A342645 T(n,n-1) = (-3n + n^2)/2.
%F A342645 T(n,n-2) = (-2n + 21n^2 - 22n^3 + 3n^4)/24.
%F A342645 T(n,n-3) = (96n - 134n^2 + 13n^3 + 37n^4 - 13n^5 + n^6)/48.
%e A342645 n\k |    0       1       2       3       4      5    6    7   8  9
%e A342645 ----+--------------------------------------------------------------
%e A342645   0 |    1;
%e A342645   1 |   -1,      1;
%e A342645   2 |   -2,     -1,      1;
%e A342645   3 |    0,     -7,      0,      1;
%e A342645   4 |    0,    -14,    -13,      2,      1;
%e A342645   5 |  120,    -46,    -65,    -15,      5,     1;
%e A342645   6 |    0,    516,   -356,   -165,     -5,     9,   1;
%e A342645   7 | 5040,   1392,    266,  -1421,   -280,    28,  14,   1;
%e A342645   8 |    0,  46320,   3772,  -5740,  -3871,  -280,  98,  20,  1;
%e A342645   9 |    0, 215280, 212724, -26272, -31437, -7791, 126, 222, 27, 1;
%e A342645 For n = 4, the polynomial that describes the 4th column of A008302 is
%e A342645 A008302(x,4) = (-14x -13x^2 +2x^3 + x^4)/4! = Sum_{j=0..4} (T(j,4)*x^j)/4!.
%t A342645 A008302T[0, 0] := 1; A008302T[-1, k_] := 0;
%t A342645 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]);
%t A342645 A342645Row[n_] := (A342645Row[n] = Expand[n!*InterpolatingPolynomial[Table[{m, A008302T[m, n]}, {m, n, 2*n + 2}], x]]);
%t A342645 A342645T[n_, k_] := Coefficient[A342645Row[n], x, k];
%Y A342645 Cf. A008302, A053657, A290030.
%K A342645 sign,tabl
%O A342645 0,4
%A A342645 _Peter Kagey_, Mar 17 2021