OFFSET
0,3
COMMENTS
The sequence of row lengths is A000041: [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...] (partition numbers p(n)).
For the ordering of this tabf array a(n,k) see Abramowitz-Stegun (A-St) ref. pp. 831-2.
This is the array n!/A117506(n,k).
For rows 1..15 of this irregular triangle see the W. Lang link.
The row sums give A263004.
The formula given below is the one obtained from the version given, e.g., in Wybourne's book for A117506(n, k). See also the Glass-Ng reference, Theorem 1, p. 701, which gives the same formula, after rewriting using also a Vandermonde determinant.
In A. Young's third paper (Q.S.A. III, see A117506), Theorem V on p. 266, CP p. 363, f/n! (the present 1/a(n,k)) appears in the decomposition of 1 for each n, that is Sum_{k = 1..p(n)} 1/a(n,k) Sum_{j=1..d(n,k)} Y'(n,k,j) = 1, with d(n,k) = A117506(n,k), and the Young operators Y' for the standard tableaux for the k-th partition of n in A-St order.
a(n,k) also appears as normalization to obtain the idempotents NP/a(n,k). See A. Young, Q.S.A. II, p. 366, CP p. 97: NP = (1/a(n,k)) (NP)^2 for each Young tableau of the shape given by the k-th partition of n in A-St order.
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
B. Wybourne, Symmetry principles and atomic spectroscopy, Wiley, New York, 1970, p. 9.
LINKS
Alois P. Heinz, Rows n = 0..30, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Kenneth Glass and Chi-Keung Ng, A Simple Proof of the Hook Length Formula, Am. Math. Monthly 111 (2004) 700 - 704.
Wolfdieter Lang, Rows 1..15.
FORMULA
a(n,k) = Product_{i=1..m(n,k)} (x_i)!/Det(x_i^(m(n,k) - j)) with the Vandermonde determinant for the variables x_i := lambda(n,k)_i + m(n,k) - i, for i, j = 1..m(n,k), where m(n,k) is the number of parts of the k-th partition of n denoted by lambda(n,k), in the A-St order (see above). Lambda(n,k)_i stands for the i-th part of the partition lambda(n,k), sorted in nonincreasing order (this is the reverse of the A-St notation for a partition).
EXAMPLE
The first rows of this irregular triangle are:
n\k 1 2 3 4 5 6 7 8 9 10 11
0: 1
1: 1
2: 2 2
3: 6 3 6
4: 24 8 12 8 24
5: 120 30 24 20 24 30 120
6: 720 144 80 144 72 45 144 72 80 144 720
...
Note that the rows are in general not symmetric.
See the W. Lang link for rows n = 1..15.
a(6,6) is related to the (self-conjugate) partition (1, 2, 3) of n = 6, taken in reverse order (3, 2, 1) with the Ferrers (or Young) diagram
_ _ _
|_|_|_| and the hook length numbers 5 3 1 ...
|_|_| 3 1
|_| 1
The product gives 5*3*1*3*1*1 = 45 = a(6,6).
MAPLE
h:= l-> (n-> mul(mul(1+l[i]-j+add(`if`(l[k]>=j, 1, 0),
k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
g:= (n, i, l)->`if`(n=0 or i=1, [h([l[], 1$n])],
`if`(i<1, [], [g(n, i-1, l)[],
`if`(i>n, [], g(n-i, i, [l[], i]))[]])):
T:= n-> g(n$2, [])[]:
seq(T(n), n=0..10); # Alois P. Heinz, Nov 05 2015
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Oct 09 2015
EXTENSIONS
Row n=0 prepended by Alois P. Heinz, Nov 05 2015
STATUS
approved