A325191 Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.

0, 0, 2, 0, 3, 3, 0, 4, 6, 4, 0, 5, 10, 10, 5, 0, 6, 15, 20, 15, 6, 0, 7, 21, 35, 35, 21, 7, 0, 8, 28, 56, 70, 56, 28, 8, 0, 9, 36, 84, 126, 126, 84, 36, 9, 0, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0, 11, 55, 165, 330, 462

Offset

0,3

Comments

The Heinz numbers of these partitions are given by A325196.

Under the Bulgarian solitaire step, these partitions form cycles of length >= 2. Length >= 2 means not the length=1 self-loop which occurs from the triangular partition when n is a triangular number. See A074909 for self-loops included. - Kevin Ryde, Sep 27 2019

Links

Table of n, a(n) for n=0..60.

FindStat, St000380: Half the perimeter of the largest rectangle that fits inside the diagram of an integer partition

FindStat, St000384: The maximal part of the shifted composition of an integer partition

FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram

Eric Weisstein's World of Mathematics, Graph Distance

Formula

Positions of zeros are A000217 = n * (n + 1) / 2.

a(n) = A074909(n) - A010054(n). - Kevin Ryde, Sep 27 2019

Example

The a(2) = 2 through a(12) = 10 partitions (empty columns not shown):
  (2)   (22)   (32)   (322)   (332)   (432)   (4322)   (4332)
  (11)  (31)   (221)  (331)   (422)   (3321)  (4331)   (4422)
        (211)  (311)  (421)   (431)   (4221)  (4421)   (4431)
                      (3211)  (3221)  (4311)  (5321)   (5322)
                              (3311)          (43211)  (5331)
                              (4211)                   (5421)
                                                       (43221)
                                                       (43311)
                                                       (44211)
                                                       (53211)

Mathematica

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

Prog

(PARI) a(n) = my(t=ceil(sqrtint(8*n+1)/2), r=n-t*(t-1)/2); if(r==0, 0, binomial(t, r)); \\ Kevin Ryde, Sep 27 2019

Crossrefs

Column k=1 of A325200.

Cf. A060687, A065770, A071724, A256617, A325166, A325169, A325178, A325179, A325181, A325187, A325188, A325189, A325195, A325196.

Sequence in context: A053727 A265208 A265020 * A209703 A279779 A279587

Adjacent sequences: A325188 A325189 A325190 * A325192 A325193 A325194

Keyword

nonn,look

Author

Gus Wiseman, Apr 11 2019

Status

approved