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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
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
Row sums are A000041. Column k = 1 is A130130. Column k = 2 is A325168.
Sequence in context: A170976 A134109 A325227 * A170978 A238353 A365676
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 11 2019
STATUS
approved