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A344607 Number of integer partitions of n with reverse-alternating sum >= 0. 61
1, 1, 2, 2, 4, 4, 8, 8, 15, 16, 27, 29, 48, 52, 81, 90, 135, 151, 220, 248, 352, 400, 553, 632, 859, 985, 1313, 1512, 1986, 2291, 2969, 3431, 4394, 5084, 6439, 7456, 9357, 10836, 13479, 15613, 19273, 22316, 27353, 31659, 38558, 44601, 53998, 62416, 75168 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

Also the number of reversed integer partitions of n with alternating sum >= 0.

A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of integer partitions of n whose parts are all even or whose greatest part is odd.

All integer partitions have alternating sum >= 0, so the non-reversed version is A000041.

Is this sequence weakly increasing? In particular, is A344611(n) <= A160786(n)?

LINKS

Table of n, a(n) for n=0..48.

FORMULA

a(n) + A344608(n) = A000041(n).

a(2n+1) = A160786(n).

EXAMPLE

The a(1) = 1 through a(8) = 15 partitions:

  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)

       (11)  (111)  (22)    (221)    (33)      (322)      (44)

                    (211)   (311)    (222)     (331)      (332)

                    (1111)  (11111)  (321)     (421)      (422)

                                     (411)     (511)      (431)

                                     (2211)    (22111)    (521)

                                     (21111)   (31111)    (611)

                                     (111111)  (1111111)  (2222)

                                                          (3311)

                                                          (22211)

                                                          (32111)

                                                          (41111)

                                                          (221111)

                                                          (2111111)

                                                          (11111111)

MATHEMATICA

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];

Table[Length[Select[IntegerPartitions[n], sats[#]>=0&]], {n, 0, 30}]

CROSSREFS

The non-reversed version is A000041.

The opposite version (rev-alt sum <= 0) is A027187, ranked by A028260.

The strict case for n > 0 is A067659 (even bisection: A344650).

The ordered version appears to be A116406 (even bisection: A114121).

The odd bisection is A160786.

The complement is counted by A344608.

The Heinz numbers of these partitions are A344609 (complement: A119899).

The even bisection is A344611.

A000070 counts partitions with alternating sum 1 (reversed: A000004).

A000097 counts partitions with alternating sum 2 (reversed: A120452).

A035363 counts partitions with alternating sum 0, ranked by A000290.

A103919 counts partitions by sum and alternating sum.

A316524 is the alternating sum of prime indices of n (reversed: A344616).

A325534/A325535 count separable/inseparable partitions.

A344610 counts partitions by sum and positive reverse-alternating sum.

A344612 counts partitions by sum and reverse-alternating sum.

A344618 gives reverse-alternating sums of standard compositions.

Cf. A006330, A071321, A071322, A124754, A239829, A239830, A344604, A344651, A344654, A344739, A344742.

Sequence in context: A262966 A034397 A200750 * A325722 A279818 A263614

Adjacent sequences:  A344604 A344605 A344606 * A344608 A344609 A344610

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 29 2021

STATUS

approved

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Last modified January 21 10:25 EST 2022. Contains 350477 sequences. (Running on oeis4.)