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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

Cf. A117506, A263004.

Sequence in context: A276484 A100641 A028421 * A081745 A240578 A273105

Adjacent sequences:  A263000 A263001 A263002 * A263004 A263005 A263006

KEYWORD

nonn,tabf,look

AUTHOR

Wolfdieter Lang, Oct 09 2015

EXTENSIONS

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

STATUS

approved

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Last modified August 16 06:38 EDT 2018. Contains 313783 sequences. (Running on oeis4.)