A263003 - OEIS

%I #36 May 08 2018 15:11:56

%S 1,1,2,2,6,3,6,24,8,12,8,24,120,30,24,20,24,30,120,720,144,80,144,72,

%T 45,144,72,80,144,720,5040,840,360,360,336,144,240,240,252,144,360,

%U 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

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

%C The sequence of row lengths is A000041: [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...] (partition numbers p(n)).

%C For the ordering of this tabf array a(n,k) see Abramowitz-Stegun (A-St) ref. pp. 831-2.

%C This is the array n!/A117506(n,k).

%C For rows 1..15 of this irregular triangle see the W. Lang link.

%C The row sums give A263004.

%C 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.

%C 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.

%C 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.

%D 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.

%D B. Wybourne, Symmetry principles and atomic spectroscopy, Wiley, New York, 1970, p. 9.

%H Alois P. Heinz, <a href="/A263003/b263003.txt">Rows n = 0..30, flattened</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Kenneth Glass and Chi-Keung Ng, <a href="http://www.jstor.org/stable/4145043?seq=1#page_scan_tab_contents">A Simple Proof of the Hook Length Formula</a>, Am. Math. Monthly 111 (2004) 700 - 704.

%H Wolfdieter Lang, <a href="/A263003/a263003.pdf">Rows 1..15.</a>

%F 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).

%e The first rows of this irregular triangle are:

%e n\k   1    2    3    4    5   6    7   8   9   10   11

%e 0:    1

%e 1:    1

%e 2:    2    2

%e 3:    6    3    6

%e 4:   24    8   12    8   24

%e 5:  120   30   24   20   24  30  120

%e 6:  720  144   80  144   72  45  144  72  80  144  720

%e ...

%e Note that the rows are in general not symmetric.

%e See the W. Lang link for rows n = 1..15.

%e 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

%e    _ _ _

%e   |_|_|_| and the hook length numbers   5  3  1 ...

%e   |_|_|                                 3  1

%e   |_|                                   1

%e The product gives 5*3*1*3*1*1 = 45 = a(6,6).

%p h:= l-> (n-> mul(mul(1+l[i]-j+add(`if`(l[k]>=j, 1, 0),

%p              k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):

%p g:= (n, i, l)->`if`(n=0 or i=1, [h([l[], 1$n])],

%p                `if`(i<1, [], [g(n, i-1, l)[],

%p                `if`(i>n, [], g(n-i, i, [l[], i]))[]])):

%p T:= n-> g(n$2, [])[]:

%p seq(T(n), n=0..10);  # _Alois P. Heinz_, Nov 05 2015

%Y Cf. A117506, A263004.

%K nonn,tabf,look

%O 0,3

%A _Wolfdieter Lang_, Oct 09 2015

%E Row n=0 prepended by _Alois P. Heinz_, Nov 05 2015