a(n,m)  tabf head (staircase) for A115198
  
  Parity of partitions of n in Abramowitz-Stegun (A-St) order.
 
  The parity (written as 1 for even (!), 0 for odd (!), because we are interested in the even partitions) of a partition of n is defined here 

  according to the parity of the number of even parts. 
 
  If a partititon has no even part it is even (zero even parts) indicated by a 1.  
  
  This is related to the parity of permutations when written in cycle notation, hence partitions. 

  E.g.: The permutation (123)(4)(5) = (123) has cycle structure <1^2,2^0,3^1,4^0,5^0> denoting the partition (3,1,1) of 5.

  The parity of this partition of 5 is obtained from the array as a(5,4) = 1 (A-St order), hence it is even ( 0 number of even parts).  

  A permutation is odd, resp. even, if the corresponding partition  is odd, resp. even.
 
  Therefore, the permutation (123)(4)(5) is even and belongs to A_5,  the (invariant) subgroup of the symmetric group S_5, called the 
  
  alternating group A_5. 


  Those even partitions which determine conjugacy classes of S_n  splitting into two equally sized classes under restriction to A_n 
  
  are denoted by '1.'. 

  This splitting occurs for those even partitions of n>=2 which have precisely one odd part or have only odd parts which are pairwise different. 
  
  See the  G. James - A. Kerber ref. given under A115200,  p. 12, 1.2.10 Lemma.

  This splitting occurs for those partitons (always even ones) which are given by the hook numbers of the main diagonal of self-conjugated 
   
  (also called self-associated) partitons of n. A self-conjugated partiton has by definition a symmetric Ferrers-Young diagram.
   
  See the G. James - A. Kerber ref. given under A115200, p. 67, 2.5.9-11.

  E.g.: n=6 has one self-conjugate partition: (3,2,1) (draw the Ferrers-Young diagram). The hook numbers for the main diagonal are 5,1, 

  hence the conjugacy class (5,1) of S_6  splits into two classes under A_6. The parity of (5,1) appears in the array as a(6,2)=1.   

  The number of elements of a conjugacy class of S_n is given by the M2 number A036039 of the corresponding partition. 
  
  See Abramowitz-Stegun pp. 831-2 for these numbers. The mentioned splitting under A_n leads to two conjugacy classes which have 

  order (M_2)/2 each. 

  The number of S_n conjugacy classes which split under A_n is given under A000700(n),n>=2.

  The number of conjugacy classes of A_n is found under A000702.  




  n\m   1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
 
  1     1  

  2     0  1 

  3     1. 0 1  
        
  4     0  1. 1  0  1

  5     1. 0  0  1  1  0  1  

  6     0  1. 1  1  0  0  0  1  1  0  1 

  7     1. 0  0  0  1  1  1  1  0  0  0  1  1  0  1  

  8     0  1. 1  1. 1  0  0  0  0  0  1  1  1  1  1  0  0  0  1  1  0  1 

  9     1. 0  0  0  0  1  1  1. 1  1  1  1  0  0  0  0  0  0  1  1  1  1  1  0  0  0  1  1  0  1

 10     0  1. 1  1. 1  1  0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1  1  0  0  0  0  0  0  0  1  1  1  1  1  0  0  0  1  1  0  1
 
   .
   .
   .
       
  n\m   1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42



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

 One could add the row for n=0 with a 1, if the part 0 is considered for n=0, and only for this n.

 For the ordering of this tabf array a(n,m) see Abramowitz-Stegun ref. pp. 831-2. 

 E.g. a(4,4) refers to the fourth partition of n=4 in this ordering, namely (1^2,2)=[1,1,2], whence a(4,4)=0 because the number of 

 even parts is odd, namely 1.  a(5,4)=1 because the partition (1,2^2)=(1,2,2) has an even number, namely 2, of even parts.



 The sequence of row sums give  A046682:  [1, 1, 2, 3, 4, 6, 8, 12, 16, 22, ...].
  

 ###################################################### eof ############################################################################