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A325350 Number of integer partitions of n whose augmented differences are weakly decreasing. 8
1, 1, 2, 3, 4, 6, 8, 10, 13, 17, 21, 26, 32, 38, 46, 56, 66, 78, 92, 106, 124, 145, 166, 191, 220, 249, 284, 325, 366, 413, 468, 523, 586, 659, 733, 817, 913, 1011, 1121, 1245, 1373, 1515, 1674, 1838, 2020, 2223, 2433, 2664, 2920, 3184, 3476, 3797, 4129, 4492 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325389.

LINKS

Table of n, a(n) for n=0..53.

FORMULA

G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(j*(j+1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 25 2019

EXAMPLE

The a(1) = 1 through a(8) = 13 partitions:

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

       (11)  (21)   (31)    (32)     (42)      (52)       (53)

             (111)  (211)   (41)     (51)      (61)       (62)

                    (1111)  (311)    (321)     (421)      (71)

                            (2111)   (411)     (511)      (521)

                            (11111)  (3111)    (3211)     (611)

                                     (21111)   (4111)     (4211)

                                     (111111)  (31111)    (5111)

                                               (211111)   (32111)

                                               (1111111)  (41111)

                                                          (311111)

                                                          (2111111)

                                                          (11111111)

For example, (4,2,1,1) has augmented differences (3,2,1,1), which are weakly decreasing, so (4,2,1,1) is counted under a(8).

MATHEMATICA

aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];

Table[Length[Select[IntegerPartitions[n], OrderedQ[Reverse[aug[#]]]&]], {n, 0, 30}]

CROSSREFS

Cf. A007294, A098859, A240026, A320466, A320509, A325349, A325353, A325354, A325356, A325357, A325358, A325361, A325364.

Sequence in context: A171997 A020702 A067996 * A027585 A123015 A005434

Adjacent sequences:  A325347 A325348 A325349 * A325351 A325352 A325353

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 23 2019

STATUS

approved

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Last modified January 18 05:09 EST 2020. Contains 330995 sequences. (Running on oeis4.)