cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Apr 11 2019

Keywords

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.

Examples

			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
		

Crossrefs

Programs

  • 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}]
  • 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

Formula

Sum_{k=1..n} k*T(n,k) = A368986(n).
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