OFFSET
0,5
COMMENTS
The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps right or down from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the maximum triangular partition contained inside the diagram.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Eric Weisstein's World of Mathematics, Graph Distance.
FORMULA
Sum_{k=1..n} k*T(n,k) = A368986(n).
EXAMPLE
Triangle begins:
1
0 1
0 2 0
0 2 1 0
0 2 3 0 0
0 2 5 0 0 0
0 2 8 1 0 0 0
0 2 9 4 0 0 0 0
0 2 12 8 0 0 0 0 0
0 2 13 15 0 0 0 0 0 0
0 2 16 23 1 0 0 0 0 0 0
0 2 17 32 5 0 0 0 0 0 0 0
0 2 20 43 12 0 0 0 0 0 0 0 0
0 2 21 54 24 0 0 0 0 0 0 0 0 0
0 2 24 67 42 0 0 0 0 0 0 0 0 0 0
0 2 25 82 66 1 0 0 0 0 0 0 0 0 0 0
MATHEMATICA
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&, Append[ptn, 0]];
Table[Length[Select[IntegerPartitions[n], otb[#]==k&]], {n, 0, 15}, {k, 0, n}]
PROG
(PARI) row(n)={my(r=vector(n+1)); forpart(p=n, my(w=#p); for(i=1, #p, w=min(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