login
A356606
Number of strict integer partitions of n where all parts have neighbors.
10
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.
Sequence in context: A111317 A105202 A240236 * A099386 A161067 A161106
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 24 2022
STATUS
approved