1,5

Also the numbers of standard Young tableaux for Young diagrams (or partitions).

The irreducible representations of S_n correspond to Young diagrams or partitions.

Partitions of n are ordered according to Abramowitz-Stegun (A-St) (see the reference, pp. 831-2). In contrast to A-St, a partition has falling parts (reverse notation of A-St).

The dimension of a representation of S_n corresponding to a Young diagram or partition is a(n,k) for the k-th partition of n in this A-St order.

The row length of this array is p(n):=A000041 (partition numbers).

One could call these numbers a(n,k) M_4 (similar to M_0, M_1, M_2, M_3 given in A111786, A036038, A036039, A036040, resp.).

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

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].

A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, pp. 831-2.

W. Lang: First 15 rows.

Eric Weisstein's World of Mathematics, Hook length formula.

a(n,k)= n!/H(n,k) with H(n,k):= product(x_i,i=1..m(n,k))/Det(x_i^(m(n,k)-j)) with the Vandermonde determinant for the variables x_i:=lambda(n,k)_i + m(n,k)-i, i,j=1..m(n,k) if m(n,k) is the number of parts of the k-th partition of n, called lambda(n,k), in the A-St order (see above). lambda(n,k)_i denotes the i-th part of the partition lambda(n,k), sorted in decreasing order (this is the reverse of the A-St notation).

a(n,k) = n!/product(h(n,k,j),j=1..n) with the hook numbers h(n,k,j) of the Young diagram of the partition lambda(n,k) in the A-St order. See the link for 'hook length formula'.

[1]; [1, 1]; [1, 2, 1]; [1, 3, 2, 3, 1]; [1, 4, 5, 6, 5, 4, 1]; [1,

5, 9, 5, 10, 16, 5, 10, 9, 5, 1];...

a(4,4)=3 because the 4th partition of n=4 in A-St order is [2,1,1],

and H(4,4)=(4!*2!*1!)/Vandermonde([4,2,1]) = (4!*2)/6 =4*2, hence

4!/H(4,4)=3.

a(4,4)=3 because the hook lengths of the Young diagram of [2,1,1]

are [4 1; 2; 1], hence 4!/(4*1*2*1)=3.

The row sums give A000085 (total number of Young tableaux with n cells).

Sequence in context: A132844 A006843 A049456 this_sequence A055089 A060117 A112592

Adjacent sequences: A117503 A117504 A117505 this_sequence A117507 A117508 A117509

nonn,easy,tabf

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 13 2006