A263003 Partition array for the products of the hook lengths of Ferrers (Young) diagrams corresponding to the partitions of n, written in Abramowitz-Stegun order.

1, 1, 2, 2, 6, 3, 6, 24, 8, 12, 8, 24, 120, 30, 24, 20, 24, 30, 120, 720, 144, 80, 144, 72, 45, 144, 72, 80, 144, 720, 5040, 840, 360, 360, 336, 144, 240, 240, 252, 144, 360, 336, 360, 840, 5040, 40320, 5760, 2016, 1440, 2880, 1920, 630, 576, 720, 960, 1152, 448, 720, 576, 2880, 1152, 630, 1440, 1920, 2016, 5760, 40320, 362880, 45360, 13440, 7560, 8640, 12960, 3456, 2240, 4320, 3024, 2160, 8640, 6480, 1920, 1680, 1680, 2160, 4320, 5184, 1920, 3024, 2240, 8640, 6480, 3456, 7560, 12960, 13440, 45360, 362880

Offset: 0

Author

Wolfdieter Lang, Oct 09 2015

Keywords

• nonn
• tabf
• look

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.

Examples

			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).
		

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.

Crossrefs

Cf. A117506, A263004.

Programs

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

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).

Extensions

Row n=0 prepended by Alois P. Heinz, Nov 05 2015