Search: id:A117506
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%I A117506
%S A117506 1,1,1,1,2,1,1,3,2,3,1,1,4,5,6,5,4,1,1,5,9,5,10,16,5,10,9,5,1,1,6,14,14,
%T A117506 15,35,21,21,20,35,14,15,14,6,1,1,7,20,28,14,21,64,70,56,42,35,90,56,70,
%U A117506 14,35,64,28,21,20,7,1
%N A117506 Dimensions of the irreducible representations of the symmetric group 
               S_n.
%C A117506 Also the numbers of standard Young tableaux for Young diagrams (or partitions).
%C A117506 The irreducible representations of S_n correspond to Young diagrams or 
               partitions.
%C A117506 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).
%C A117506 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.
%C A117506 The row length of this array is p(n):=A000041 (partition numbers).
%C A117506 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.).
%D A117506 G. B. Wybourne, Symmetry principles and atomic spectroscopy, Wiley, New 
               York, 1970, p. 9.
%H A117506 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A117506 A. M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/
               Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</
               a>, pp. 831-2.
%H A117506 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A117506.text">
               First 15 rows.</a>
%H A117506 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HookLengthFormula.html">Hook length formula.</a>
%F A117506 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).
%F A117506 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'.
%e A117506 [1]; [1, 1]; [1, 2, 1]; [1, 3, 2, 3, 1]; [1, 4, 5, 6, 5, 4, 1]; [1,
%e A117506 5, 9, 5, 10, 16, 5, 10, 9, 5, 1];...
%e A117506 a(4,4)=3 because the 4th partition of n=4 in A-St order is [2,1,1],
%e A117506 and H(4,4)=(4!*2!*1!)/Vandermonde([4,2,1]) = (4!*2)/6 =4*2, hence
%e A117506 4!/H(4,4)=3.
%e A117506 a(4,4)=3 because the hook lengths of the Young diagram of [2,1,1]
%e A117506 are [4 1; 2; 1], hence 4!/(4*1*2*1)=3.
%Y A117506 The row sums give A000085 (total number of Young tableaux with n cells).
%Y A117506 Sequence in context: A132844 A006843 A049456 this_sequence A055089 A060117 
               A112592
%Y A117506 Adjacent sequences: A117503 A117504 A117505 this_sequence A117507 A117508 
               A117509
%K A117506 nonn,easy,tabf
%O A117506 1,5
%A A117506 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 13 
               2006

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