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A115201
Number of even parts of partitions of n in the Abramowitz-Stegun (A-St) order.
1
0, 1, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 1, 0, 2, 1, 0, 1, 0, 2, 0, 1, 1, 3, 0, 2, 1, 0, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 3, 0, 2, 1, 0, 1, 0, 2, 0, 2, 1, 1, 1, 3, 1, 0, 2, 0, 2, 4, 1, 1, 3, 0, 2, 1, 0, 0, 1, 1, 1, 1, 0, 2, 0, 2, 2, 2, 0, 1, 1, 1
OFFSET
0,9
COMMENTS
A conjugacy class of the symmetric group S_n with the cycle structure given by the partition, listed in the A-St order, consists of even, resp. odd, permutations if a(n,m) is even, resp. odd.
See A115198 for the parity of a(n,m) with 1 for even, 0 for odd (main entry).
See A115199 for the parity of a(n,m) with 0 for even, 1 for odd.
The parity of these numbers determines whether a conjugacy class of the symmetric group S_n, which is determined by its cycle structure, consists of even or odd permutations.
The row length sequence of this triangle is p(n)=A000041(n) (number of partitions).
FORMULA
a(n,m) = Sum_{j=1..floor(n/2)} e(n,m,2*j) with the exponents e(n,m,k) of the m-th partition of n in the A-St order; i.e. the sum of the exponents of the even parts of the partition (1^e(n,m,1),2^e(n,m,2),..., n^e(n,m,n)).
EXAMPLE
[0];[1, 0];[0, 1, 0];[1, 0, 2, 1, 0];[0, 1, 1, 0, 2, 1, 0];...
CROSSREFS
The sequence of row lengths is A066898 (total number of even parts in all partitions of n).
Sequence in context: A303942 A321928 A321917 * A354100 A118229 A172250
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Feb 23 2006
STATUS
approved