OFFSET
0,5
COMMENTS
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.
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.
For the specified value of n, the second Maple program lists the partitions of n counted by a(n).
Number of integer partitions of n with the same number of odd parts as their conjugate. - Gus Wiseman, Jun 27 2021
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
EXAMPLE
a(3) = 1 because we have [2,1]. The partitions [3] and [1,1,1] do not qualify.
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.
MAPLE
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);
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;
# alternative Maple program:
b:= proc(n, i, s, t) option remember; `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-irem(i, 2), -t))))
end:
a:= n-> b(n$2, 0, 1):
seq(a(n), n=0..60); # Alois P. Heinz, Oct 19 2016
MATHEMATICA
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 *)
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 *)
CROSSREFS
Comparing even parts to odd conjugate parts gives A277579.
Comparing product of parts to product of conjugate parts gives A325039.
The reverse version is A345196.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 18 2016
STATUS
approved