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A325189
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Regular triangle read by rows where T(n,k) is the number of integer partitions of n with maximum origin-to-boundary graph-distance equal to k.
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16
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1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 0, 3, 2, 0, 0, 0, 3, 2, 2, 0, 0, 0, 1, 6, 2, 2, 0, 0, 0, 0, 7, 4, 2, 2, 0, 0, 0, 0, 6, 8, 4, 2, 2, 0, 0, 0, 0, 4, 12, 6, 4, 2, 2, 0, 0, 0, 0, 1, 15, 12, 6, 4, 2, 2, 0, 0, 0, 0, 0, 17, 15, 10, 6, 4, 2, 2
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OFFSET
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0,6
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COMMENTS
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The maximum origin-to-boundary graph-distance of an integer partition is one plus the maximum number of unit steps East or South in the Young diagram that can be followed, starting from the upper-left square, to reach a boundary square in the lower-right quadrant. It is also the side-length of the minimum triangular partition containing the diagram.
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
1
0 1
0 0 2
0 0 1 2
0 0 0 3 2
0 0 0 3 2 2
0 0 0 1 6 2 2
0 0 0 0 7 4 2 2
0 0 0 0 6 8 4 2 2
0 0 0 0 4 12 6 4 2 2
0 0 0 0 1 15 12 6 4 2 2
0 0 0 0 0 17 15 10 6 4 2 2
0 0 0 0 0 14 23 16 10 6 4 2 2
0 0 0 0 0 10 30 23 14 10 6 4 2 2
0 0 0 0 0 5 39 29 24 14 10 6 4 2 2
0 0 0 0 0 1 42 42 31 22 14 10 6 4 2 2
Row 9 counts the following partitions:
(432) (54) (63) (72) (81) (9)
(3321) (333) (621) (711) (21111111) (111111111)
(4221) (441) (6111) (2211111)
(4311) (522) (222111) (3111111)
(531) (321111)
(3222) (411111)
(5211)
(22221)
(32211)
(33111)
(42111)
(51111)
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MATHEMATICA
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otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&, Append[ptn, 0]];
Table[Length[Select[IntegerPartitions[n], otbmax[#]==k&]], {n, 0, 15}, {k, 0, n}]
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PROG
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(PARI) row(n)={my(r=vector(n+1)); forpart(p=n, my(w=0); for(i=1, #p, w=max(w, #p-i+p[i])); r[w+1]++); r} \\ Andrew Howroyd, Jan 12 2024
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CROSSREFS
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Cf. A065770, A096771, A115720, A115994, A139582, A325169, A325183, A325188, A325195, A325200, A366157.
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KEYWORD
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AUTHOR
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STATUS
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approved
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