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A339737
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Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.
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8
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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, 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, 3, 6, 8, 9, 6, 5, 3, 2, 1, 1, 0, 15, 3, 8, 11, 11, 10, 7, 5, 3, 2, 1, 1, 0
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OFFSET
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0,7
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COMMENTS
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We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.
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LINKS
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EXAMPLE
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Triangle begins:
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 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 3 6 8 9 6 5 3 2 1 1 0
15 3 8 11 11 10 7 5 3 2 1 1 0
18 4 9 13 15 13 10 7 5 3 2 1 1 0
22 5 10 17 19 18 14 11 7 5 3 2 1 1 0
27 5 13 20 24 23 20 14 11 7 5 3 2 1 1 0
For example, row n = 9 counts the following partitions:
(3321) (432) (333) (54) (522) (63) (72) (81) (9)
(22221) (3222) (4311) (441) (531) (621) (711)
(32211) (33111) (4221) (5211) (6111)
(222111) (3111111) (42111) (51111)
(321111) (411111)
(2211111)
(21111111)
(111111111)
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MATHEMATICA
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maxgap[q_]:=Max@@Complement[Range[0, If[q=={}, 0, Max[q]]], q];
Table[Length[Select[IntegerPartitions[n], maxgap[#]==k&]], {n, 0, 15}, {k, 0, n}]
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PROG
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(PARI)
S(n, k)={if(k>n, O(x*x^n), x^k*(S(n-k, k+1) + 1)/(1 - x^k))}
ColGf(k, n) = {(k==0) + S(n, k+1)/prod(j=1, k-1, 1 - x^j + O(x^max(1, n-k)))}
A(n, m=n)={Mat(vector(m+1, k, Col(ColGf(k-1, n), -(n+1))))}
{ my(M=A(10)); for(i=1, #M, print(M[i, 1..i])) } \\ Andrew Howroyd, Jan 13 2024
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CROSSREFS
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The version for least gap is A264401, with Heinz number encoding A257993.
An encoding (of greatest gap) using Heinz numbers is A339662.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A048004 counts compositions by greatest part.
A064428 counts partitions of nonnegative crank.
A073491 list numbers with gap-free prime indices.
A107428 counts gap-free compositions.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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