OFFSET
0,6
COMMENTS
In the first Maple program (improvable) AS gives the positive alternating sum of a finite sequence s, EP gives the number of even 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).
Also the number of integer partitions of n with as many even parts as odd parts in the conjugate partition. - Gus Wiseman, Jul 26 2021
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
EXAMPLE
a(9) = 6: [2,1,1,1,1,1,1,1], [3,2,1,1,1,1], [3,3,2,1], [4,2,2,1], [4,3,1,1], [5,4].
a(10) = 7: [1,1,1,1,1,1,1,1,1,1], [3,2,2,1,1,1], [3,3,1,1,1,1], [4,2,1,1,1,1], [4,3,2,1], [5,5], [6,4].
a(11) = 9: [2,1,1,1,1,1,1,1,1,1], [3,2,1,1,1,1,1,1], [3,3,2,1,1,1], [3,3,3,2], [4,2,2,1,1,1], [4,3,1,1,1,1], [5,2,2,2], [5,4,1,1], [6,5].
MAPLE
with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: EP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 0 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]) = EP(P[k]) then c := c+1 else end if end do: c end proc: seq(a(n), n = 0 .. 30);
n := 8: with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: EP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 0 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]) = EP(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+1, 2), -t))))
end:
a:= n-> b(n$2, 0, 1):
seq(a(n), n=0..60);
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+1, 2], -t]]]]; a[n_] := b[n, n, 0, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 21 2016, translated from Maple *)
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]]; Table[Length[Select[IntegerPartitions[n], Count[#, _?EvenQ]==Count[conj[#], _?OddQ]&]], {n, 0, 15}] (* Gus Wiseman, Jul 26 2021 *)
PROG
(Sage)
def a(n):
AS = lambda s: abs(sum((-1)^i*t for i, t in enumerate(s)))
EP = lambda s: sum((t+1)%2 for t in s)
return sum(AS(p) == EP(p) for p in Partitions(n))
print([a(n) for n in (0..30)]) # Peter Luschny, Oct 21 2016
CROSSREFS
Comparing odd parts to odd conjugate parts gives A277103.
Comparing product of parts to product of conjugate parts gives A325039.
Comparing the rev-alt sum to that of the conjugate gives 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 and Alois P. Heinz, Oct 20 2016
STATUS
approved