%I #14 Aug 29 2019 16:19:02

%S 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,

%T 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,

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

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

%C The array with 0 and 1 interchanged is A115199.

%C The partitions appear in the Abramowitz-Stegun (A-St) order (see the reference, pp. 831-2).

%C 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.

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

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972.

%H W. Lang: <a href="/A115198/a115198.txt">First 10 rows.</a>

%F 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.

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

%e a(4,4)=0 because it refers to the 4th partition of n=4 of the

%e mentioned A-St ordering, namely to (1^2,2^1)=(1,1,2) which has an odd number

%e (1) of even parts.

%e a(5,4)=1 because (1^1,2^2)=(1,2,2) has an even number of even parts

%e (the number of even parts is in fact 2).

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

%K nonn,easy,tabf

%O 0,1

%A _Wolfdieter Lang_, Feb 23 2006