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Number of integer partitions of n with origin-to-boundary graph-distance equal to 2.
7

%I #12 Jan 08 2021 21:16:59

%S 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,

%T 45,48,49,52,53,56,57,60,61,64,65,68,69,72,73,76,77,80,81,84,85,88,89,

%U 92,93,96,97,100,101,104,105,108,109,112,113,116,117,120,121

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

%C 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.

%H Colin Barker, <a href="/A325168/b325168.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F From _Colin Barker_, Apr 08 2019: (Start)

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

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

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

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

%F (End)

%e The a(3) = 1 through a(10) = 16 partitions:

%e (21) (22) (32) (33) (43) (44) (54) (55)

%e (31) (41) (42) (52) (53) (63) (64)

%e (211) (221) (51) (61) (62) (72) (73)

%e (311) (222) (511) (71) (81) (82)

%e (2111) (411) (2221) (611) (711) (91)

%e (2211) (4111) (2222) (6111) (811)

%e (3111) (22111) (5111) (22221) (7111)

%e (21111) (31111) (22211) (51111) (22222)

%e (211111) (41111) (222111) (61111)

%e (221111) (411111) (222211)

%e (311111) (2211111) (511111)

%e (2111111) (3111111) (2221111)

%e (21111111) (4111111)

%e (22111111)

%e (31111111)

%e (211111111)

%t otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];

%t Table[Length[Select[IntegerPartitions[n],otb[#]==2&]],{n,0,30}]

%o (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

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

%K nonn,easy

%O 0,5

%A _Gus Wiseman_, Apr 05 2019