A226167 Array read by antidiagonals: a(i,j) is the number of ways of labeling a tableau of shape (i,1^j) with the integers 1, 2, ... i+j-2 (each label being used once) such that the first row is decreasing, and the first column has m-1 labels.

1, 3, 1, 12, 5, 1, 60, 27, 7, 1, 360, 168, 48, 9, 1, 2520, 1200, 360, 75, 11, 1, 20160, 9720, 3000, 660, 108, 13, 1, 181440, 88200, 27720, 6300, 1092, 147, 15, 1, 1814400, 887040, 282240, 65520, 11760, 1680, 192, 17, 1, 19958400, 9797760, 3144960, 740880, 136080, 20160, 2448, 243, 19, 1

Offset: 1

Author

John M. Campbell, May 29 2013

Keywords

• nonn
• tabl

Comments

For an arbitrary composition c, let F_c^p denote the linear transformation of NSym that is adjoint to multiplication by the fundamental quasi-symmetric function indexed by c. Then a(i,j) equals the coefficient of H_(1,1) in (F_(1)^p)^(i+j-2)(H_(i,1^j)) (see below SAGE program, and Corollary 2.7 in the below link).

Let M(n) = [a(i,j)]_{n x n}. Then det(M(n))=A000178(n)=the n-th superfactorial.

Let p_n(x) denote the polynomial such that a(x,n)=p_n(x). Then the coefficient of x in p_n(x) is |A009575(n)|. For example, p_4(x)=4x^3+18x^2+26x+12, and the coefficient of x in p_4(x) is |A009575(4)|=26.

First row is A001710. Second row is A138772. Fourth row is A136659.

Examples

			There are a(3,2) = 7 ways of labeling the tableau of shape (3,1,1) with 1, 2 and 3 (with each label being used once) such that the first row is decreasing and the first column has 1 label:
1    2    3    X    X    X    X
X    X    X    1    2    3    X
X32  X31  X21  X32  X31  X21  321
The matrix [a(i,j)]_(6 x 6) is given below:
[1  3  12   60   360   2520]
[1  5  27  168  1200   9720]
[1  7  48  360  3000  27720]
[1  9  75  660  6300  65520]
[1 11 108 1092 11760 136080]
[1 13 147 1680 20160 257040]
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• C. Berg, N. Bergeron, F. Saliola, L. Serrano, and M. Zabrocki, A Lift of the Schur and Hall-Littlewood Bases to Non-Commutative Symmetric Functions, 10-11.

Crossrefs

Cf. A000178, A009575.

Main diagonal gives: A023999. - Alois P. Heinz, Jan 21 2014

Programs

Maple

a:= (i, j)-> (i+j-2)!/i!*(2*i+j-1)*j/2:
seq(seq(a(i, 1+d-i), i=1..d), d=1..12);  # Alois P. Heinz, Jan 21 2014

Mathematica

a[n_,k_]:=(n+k-2)!/n!*(2*n+k-1)*k/2 ;
Print[Array[a[#1,#2]&,{50,50}]//MatrixForm]
(* A program which gives a list of tableaux *)
a[i_, j_] :=  Module[{f, list1, el, emptylist, n},
  f[q_] := StringReplace[StringReplace[StringReplace[    StringReplace[ToString[q], ToString[i + j - 1] -> "X"], ", " -> ""], "{" -> ""], "}" -> ""]; list1 = Permutations[Join[Table[q, {q, 1, i + j - 2}], {i + j - 1, i + j - 1}]]; el[q_] := First[Take[list1, {q, q}]]; emptylist = {}; n = 1; While[n < 1 + Length[list1], If[Take[el[n], {j + 1, i + j}] == Sort[Take[el[n], {j + 1, i + j}], Greater] && Count[Take[el[n], {1, j + 1}], i + j - 1] == 2, emptylist = Append[emptylist, f[el[n]]], Null]; n++]; Print[emptylist]]

Sage

NSym = NonCommutativeSymmetricFunctions(QQ) ;
QSym = QuasiSymmetricFunctions(QQ) ;
F = QSym.Fundamental() ;
H = NSym.complete() ;
def a(n, m):
     expr = H([n]+[1 for q in range(m)]) ;
     w=1 ;
     while w
				

Formula

a(i,j) = (i+j-2)!/i!*(2*i+j-1)*j/2.