|
|
A325188
|
|
Regular triangle read by rows where T(n,k) is the number of integer partitions of n with origin-to-boundary graph-distance equal to k.
|
|
14
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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
|
Cf. A000245, A065770, A096771, A115994, A325169, A325183, A325187, A325189, A325191, A325195, A325200, A368986.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|