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A277103 Number of partitions of n for which the number of odd parts is equal to the positive alternating sum of the parts.

%I A277103 #25 Jul 06 2021 02:52:23
%S A277103 1,1,0,1,3,3,1,3,10,10,4,10,27,27,13,28,69,69,37,72,161,162,96,171,
%T A277103 361,364,230,388,768,777,522,836,1581,1605,1128,1739,3145,3203,2345,
%U A277103 3495,6094,6225,4712,6831,11511,11794,9198,13010,21293,21875,17496,24239
%N A277103 Number of partitions of n for which the number of odd parts is equal to the positive alternating sum of the parts.
%C A277103 It follows by conjugation that the partition statistics "alternating sum" and "number of odd parts" are equidistributed. Consequently, the self-conjugate partitions satisfy the required condition.
%C A277103 In the first Maple program (improvable) AS gives the positive alternating sum of a finite sequence s, OP gives the number of odd terms of a finite sequence of positive integers.
%C A277103 For the specified value of n, the second Maple program lists the partitions of n counted by a(n).
%C A277103 Number of integer partitions of n with the same number of odd parts as their conjugate. - _Gus Wiseman_, Jun 27 2021
%H A277103 Alois P. Heinz, <a href="/A277103/b277103.txt">Table of n, a(n) for n = 0..1000</a>
%e A277103 a(3) = 1 because we have [2,1]. The partitions [3] and [1,1,1] do not qualify.
%e A277103 a(4) = 3 because we have [3,1], [2,2], and [2,1,1]. The partitions [4] and [1,1,1,1] do not qualify.
%p A277103 with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: OP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 1 then ct := ct+1 else  end if end do: ct end proc: a := proc (n) local P, c, k: P := partition(n): c := 0: for k to nops(P) do if AS(P[k]) = OP(P[k]) then c := c+1 else end if end do: c end proc: seq(a(n), n = 0 .. 50);
%p A277103 n := 8: with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: OP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 1 then ct := ct+1 else  end if end do: ct end proc: P := partition(n): C := {}: for k to nops(P) do if AS(P[k]) = OP(P[k]) then C := `union`(C, {P[k]}) else  end if end do: C;
%p A277103 # alternative Maple program:
%p A277103 b:= proc(n, i, s, t) option remember; `if`(n=0,
%p A277103       `if`(s=0, 1, 0), `if`(i<1, 0, b(n, i-1, s, t)+
%p A277103       `if`(i>n, 0, b(n-i, i, s+t*i-irem(i, 2), -t))))
%p A277103     end:
%p A277103 a:= n-> b(n$2, 0, 1):
%p A277103 seq(a(n), n=0..60);  # _Alois P. Heinz_, Oct 19 2016
%t A277103 b[n_, i_, s_, t_] := b[n, i, s, t] = If[n == 0, If[s == 0, 1, 0], If[i<1, 0, b[n, i-1, s, t] + If[i>n, 0, b[n-i, i, s + t*i - Mod[i, 2], -t]]]]; a[n_] := b[n, n, 0, 1]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Dec 21 2016, after _Alois P. Heinz_ *)
%t A277103 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]]; Table[Length[Select[IntegerPartitions[n],Count[#,_?OddQ]==Count[conj[#],_?OddQ]&]],{n,0,15}] (* _Gus Wiseman_, Jun 27 2021 *)
%Y A277103 Comparing even parts to odd conjugate parts gives A277579.
%Y A277103 Comparing product of parts to product of conjugate parts gives A325039.
%Y A277103 The reverse version is A345196.
%Y A277103 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A277103 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A277103 A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
%Y A277103 A124754 gives alternating sums of standard compositions (reverse: A344618).
%Y A277103 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A277103 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A277103 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A277103 Cf. A000070, A000097, A000700, A006330, A027187, A027193, A236559, A257991, A325534, A325535, A344607, A344651.
%K A277103 nonn
%O A277103 0,5
%A A277103 _Emeric Deutsch_, Oct 18 2016