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.

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.

Cf. A000245, A065770, A096771, A115994, A325169, A325183, A325187, A325189, A325191, A325195, A325200, A368986.

Sequence in context: A170976 A134109 A325227 * A170978 A238353 A365676

Adjacent sequences: A325185 A325186 A325187 * A325189 A325190 A325191

Keyword

nonn,tabl

Author

Gus Wiseman, Apr 11 2019

Status

approved