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A226167
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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.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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a(i,j) = (i+j-2)!/i!*(2*i+j-1)*j/2.
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EXAMPLE
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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]
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MAPLE
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a:= (i, j)-> (i+j-2)!/i!*(2*i+j-1)*j/2:
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MATHEMATICA
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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]]
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PROG
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(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<n+m-1:
expr = expr.skew_by(F[1])
w+=1
return(expr.coefficients()[0])
print(matrix([[a(j+1, i+1) for i in range(7)] for j in range (7)]))
list1=[] ;
n=0 ;
while n<10:
list1 = list1 + [a(i+1, n+1-i) for i in range(n+1)]
n+=1
print(list1)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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