A356606 Number of strict integer partitions of n where all parts have neighbors.

1, 0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 2, 3, 2, 2, 5, 2, 4, 5, 5, 4, 8, 5, 7, 9, 8, 8, 13, 10, 11, 16, 13, 15, 20, 18, 18, 27, 21, 26, 31, 30, 30, 43, 34, 42, 49, 48, 48, 65, 56, 65, 76, 74, 77, 97, 88, 98, 117, 111, 119, 143, 137, 146, 175, 165, 182, 208

Offset

0,10

Comments

A part x has a neighbor if either x - 1 or x + 1 is a part.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 301 terms from John Tyler Rascoe)

John Tyler Rascoe, Python program

Formula

G.f.: 1 + Sum_{i>0} A(x,i), where A(x,i) = x^((2*i)+1) * G(x,i+1) for i > 0, is the g.f. for partitions of this kind with least part i, and G(x,k) = 1 + x^(k+1) * G(x,k+1) + Sum_{m>=0} x^(2*(k+m)+5) * G(x,m+k+3). - John Tyler Rascoe, Feb 16 2024

Example

The a(n) partitions for n = 0, 1, 3, 9, 15, 18, 20, 24 (A = 10, B = 11):
  ()  .  (21)  (54)   (87)     (765)    (7643)   (987)
               (432)  (654)    (6543)   (8732)   (8754)
                      (54321)  (7632)   (9821)   (9843)
                               (8721)   (65432)  (A932)
                               (65421)           (BA21)
                                                 (87432)
                                                 (87621)
                                                 (765321)

Mathematica

Table[Length[Select[IntegerPartitions[n], Function[ptn, UnsameQ@@ptn&&And@@Table[MemberQ[ptn, x-1]||MemberQ[ptn, x+1], {x, Union[ptn]}]]]], {n, 0, 30}]

Prog

(Python) # see linked program

Crossrefs

This is the strict case of A355393 and A355394.

The complement is counted by A356607, non-strict A356235 and A356236.

A000041 counts integer partitions, strict A000009.

A000837 counts relatively prime partitions, ranked by A289509.

A007690 counts partitions with no singletons, complement A183558.

Cf. A137921, A325160, A328171, A328172, A328187, A328220, A328221, A356237.

Sequence in context: A111317 A105202 A240236 * A099386 A161067 A161106

Adjacent sequences: A356603 A356604 A356605 * A356607 A356608 A356609

Keyword

nonn

Author

Gus Wiseman, Aug 24 2022

Status

approved