A115199 -id:A115199 - cp's OEIS frontend

A115198 Parity of partitions of n, with 1 for even, 0 for odd (!). The definition follows.

1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0

Offset: 0

Wolfdieter Lang, Feb 23 2006

The array with 0 and 1 interchanged is A115199.
The partitions appear in the Abramowitz-Stegun (A-St) order (see the reference, pp. 831-2).
A partition of n is (here) called even, resp. odd, if the number of even parts is even, resp. odd. A partition with no (0) even part is therefore even. Because the parity of permutations is linked, via their cycle structure, to the number of even parts of partitions one uses here 1 in order to mark the relevant (even) partitions.
The row length sequence of this array is p(n)=A000041(n) (number of partitions).

			[1];[0,1];[1,0,1];[0,1,1,0,1];[1,0,0,1,1,0,1];...
a(4,4)=0 because it refers to the 4th partition of n=4 of the
mentioned A-St ordering, namely to (1^2,2^1)=(1,1,2) which has an odd number
(1) of even parts.
a(5,4)=1 because (1^1,2^2)=(1,2,2) has an even number of even parts
(the number of even parts is in fact 2).

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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972.
W. Lang: First 10 rows.

The sequence of row lengths is A046682 (number of cycle types for even permutations).

a(n,m)= 1 if sum(e(n,m,2*j),j=1..floor(n/2)) is even, else 0, 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)) is even iff a(n,m)=1.

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

Original entry on oeis.org

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

Wolfdieter Lang, Feb 23 2006

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).

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

W. Lang: First 10 rows.

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

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)).

cp's OEIS Frontend

A115198 Parity of partitions of n, with 1 for even, 0 for odd (!). The definition follows.

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