A325168 Number of integer partitions of n with origin-to-boundary graph-distance equal to 2.

0, 0, 0, 1, 3, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 97, 100, 101, 104, 105, 108, 109, 112, 113, 116, 117, 120, 121

Offset

0,5

Comments

The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps left or down from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the maximum triangular partition contained inside it.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Formula

From Colin Barker, Apr 08 2019: (Start)

G.f.: x^3*(1 + 2*x + x^2 + x^3 - x^4) / ((1 - x)^2*(1 + x)).

a(n) = a(n-1) + a(n-2) - a(n-3) for n>7.

a(n) = 2*n - 4 for n>4 and even.

a(n) = 2*n - 5 for n>4 and odd.

(End)

Example

The a(3) = 1 through a(10) = 16 partitions:
  (21)  (22)   (32)    (33)     (43)      (44)       (54)        (55)
        (31)   (41)    (42)     (52)      (53)       (63)        (64)
        (211)  (221)   (51)     (61)      (62)       (72)        (73)
               (311)   (222)    (511)     (71)       (81)        (82)
               (2111)  (411)    (2221)    (611)      (711)       (91)
                       (2211)   (4111)    (2222)     (6111)      (811)
                       (3111)   (22111)   (5111)     (22221)     (7111)
                       (21111)  (31111)   (22211)    (51111)     (22222)
                                (211111)  (41111)    (222111)    (61111)
                                          (221111)   (411111)    (222211)
                                          (311111)   (2211111)   (511111)
                                          (2111111)  (3111111)   (2221111)
                                                     (21111111)  (4111111)
                                                                 (22111111)
                                                                 (31111111)
                                                                 (211111111)

Mathematica

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&, Append[ptn, 0]];
Table[Length[Select[IntegerPartitions[n], otb[#]==2&]], {n, 0, 30}]

Prog

(PARI) concat([0, 0, 0], Vec(x^3*(1 + 2*x + x^2 + x^3 - x^4) / ((1 - x)^2*(1 + x)) + O(x^80))) \\ Colin Barker, Apr 08 2019

Crossrefs

Cf. A006918, A065770, A115994, A117485, A257990, A297113, A325135, A325166, A325169, A325170.

Sequence in context: A189032 A189130 A189213 * A211703 A274430 A107792

Adjacent sequences: A325165 A325166 A325167 * A325169 A325170 A325171

Keyword

nonn,easy

Author

Gus Wiseman, Apr 05 2019

Status

approved