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A325189
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.
16
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
OFFSET
0,6
COMMENTS
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Bridget Eileen Tenner, Reduced word manipulation: patterns and enumeration, J. Algebr. Comb. 46, No. 1, 189-217 (2017), table 1.
Tewodros Amdeberhan, George E. Andrews, and Cristina Ballantine, Hook length and symplectic content in partitions, arXiv:2205.07322 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Graph Distance
FORMULA
Sum_{k=1..n} k*T(n,k) = A366157(n). - Andrew Howroyd, Jan 12 2024
EXAMPLE
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)
MATHEMATICA
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&, Append[ptn, 0]];
Table[Length[Select[IntegerPartitions[n], otbmax[#]==k&]], {n, 0, 15}, {k, 0, n}]
PROG
(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
CROSSREFS
Row sums are A000041. Column sums are A071724.
Sequence in context: A346632 A230595 A345957 * A357637 A130731 A287240
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 11 2019
STATUS
approved