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
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 11 2019
STATUS
approved