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A115201 Number of even parts of partitions of n in the Abramowitz-Stegun (A-St) order. 1

%I #16 Jun 08 2022 16:29:56

%S 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,

%T 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,

%U 1,1,1,0,2,0,2,2,2,0,1,1,1

%N Number of even parts of partitions of n in the Abramowitz-Stegun (A-St) order.

%C 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.

%C See A115198 for the parity of a(n,m) with 1 for even, 0 for odd (main entry).

%C See A115199 for the parity of a(n,m) with 0 for even, 1 for odd.

%C 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.

%C The row length sequence of this triangle is p(n)=A000041(n) (number of partitions).

%H W. Lang: <a href="/A115201/a115201.txt">First 10 rows.</a>

%F 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)).

%e [0];[1, 0];[0, 1, 0];[1, 0, 2, 1, 0];[0, 1, 1, 0, 2, 1, 0];...

%Y The sequence of row lengths is A066898 (total number of even parts in all partitions of n).

%K nonn,easy,tabf

%O 0,9

%A _Wolfdieter Lang_, Feb 23 2006

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Last modified March 28 16:00 EDT 2024. Contains 371254 sequences. (Running on oeis4.)