%I #17 Jan 14 2024 11:50:22

%S 1,1,0,1,1,0,2,0,1,0,2,1,1,1,0,3,1,1,1,1,0,4,1,2,2,1,1,0,5,1,3,2,2,1,

%T 1,0,6,2,3,4,3,2,1,1,0,8,2,4,5,4,3,2,1,1,0,10,2,5,7,6,5,3,2,1,1,0,12,

%U 3,6,8,9,6,5,3,2,1,1,0,15,3,8,11,11,10,7,5,3,2,1,1,0

%N Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.

%C We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.

%H Andrew Howroyd, <a href="/A339737/b339737.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%H George E. Andrews and David Newman, <a href="https://doi.org/10.1007/s00026-019-00427-w">Partitions and the Minimal Excludant</a>, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.

%H Brian Hopkins, James A. Sellers, and Dennis Stanton, <a href="https://arxiv.org/abs/2009.10873">Dyson's Crank and the Mex of Integer Partitions</a>, arXiv:2009.10873 [math.CO], 2020.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Mex_(mathematics)">Mex (mathematics)</a>

%e Triangle begins:

%e 1

%e 1 0

%e 1 1 0

%e 2 0 1 0

%e 2 1 1 1 0

%e 3 1 1 1 1 0

%e 4 1 2 2 1 1 0

%e 5 1 3 2 2 1 1 0

%e 6 2 3 4 3 2 1 1 0

%e 8 2 4 5 4 3 2 1 1 0

%e 10 2 5 7 6 5 3 2 1 1 0

%e 12 3 6 8 9 6 5 3 2 1 1 0

%e 15 3 8 11 11 10 7 5 3 2 1 1 0

%e 18 4 9 13 15 13 10 7 5 3 2 1 1 0

%e 22 5 10 17 19 18 14 11 7 5 3 2 1 1 0

%e 27 5 13 20 24 23 20 14 11 7 5 3 2 1 1 0

%e For example, row n = 9 counts the following partitions:

%e (3321) (432) (333) (54) (522) (63) (72) (81) (9)

%e (22221) (3222) (4311) (441) (531) (621) (711)

%e (32211) (33111) (4221) (5211) (6111)

%e (222111) (3111111) (42111) (51111)

%e (321111) (411111)

%e (2211111)

%e (21111111)

%e (111111111)

%t maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];

%t Table[Length[Select[IntegerPartitions[n],maxgap[#]==k&]],{n,0,15},{k,0,n}]

%o (PARI)

%o S(n,k)={if(k>n, O(x*x^n), x^k*(S(n-k,k+1) + 1)/(1 - x^k))}

%o ColGf(k,n) = {(k==0) + S(n,k+1)/prod(j=1, k-1, 1 - x^j + O(x^max(1,n-k)))}

%o A(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}

%o { my(M=A(10)); for(i=1, #M, print(M[i,1..i])) } \\ _Andrew Howroyd_, Jan 13 2024

%Y Column k = 0 is A000009.

%Y Row sums are A000041.

%Y Central diagonal is A000041.

%Y Column k = 1 is A087897.

%Y The version for least gap is A264401, with Heinz number encoding A257993.

%Y The version for greatest difference is A286469 or A286470.

%Y An encoding (of greatest gap) using Heinz numbers is A339662.

%Y A000070 counts partitions with a selected part.

%Y A006128 counts partitions with a selected position.

%Y A015723 counts strict partitions with a selected part.

%Y A048004 counts compositions by greatest part.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A064391 is the version for crank.

%Y A064428 counts partitions of nonnegative crank.

%Y A073491 list numbers with gap-free prime indices.

%Y A107428 counts gap-free compositions.

%Y A238709/A238710 counts partitions by least/greatest difference.

%Y A342050/A342051 have prime indices with odd/even least gap.

%Y Cf. A001223, A002110, A018818, A063250, A088860, A098743, A279945.

%K nonn,tabl

%O 0,7

%A _Gus Wiseman_, Apr 20 2021

%E Offset corrected by _Andrew Howroyd_, Jan 13 2024