A325189 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with maximum origin-to-boundary graph-distance equal to k.

1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 0, 3, 2, 0, 0, 0, 3, 2, 2, 0, 0, 0, 1, 6, 2, 2, 0, 0, 0, 0, 7, 4, 2, 2, 0, 0, 0, 0, 6, 8, 4, 2, 2, 0, 0, 0, 0, 4, 12, 6, 4, 2, 2, 0, 0, 0, 0, 1, 15, 12, 6, 4, 2, 2, 0, 0, 0, 0, 0, 17, 15, 10, 6, 4, 2, 2

Offset: 0

Author

Gus Wiseman, Apr 11 2019

Keywords

• nonn
• tabl

Comments

The maximum origin-to-boundary graph-distance of an integer partition is one plus the maximum number of unit steps East or South in the Young diagram that can be followed, starting from the upper-left square, to reach a boundary square in the lower-right quadrant. It is also the side-length of the minimum triangular partition containing the diagram.

Examples

			Triangle begins:
  1
  0  1
  0  0  2
  0  0  1  2
  0  0  0  3  2
  0  0  0  3  2  2
  0  0  0  1  6  2  2
  0  0  0  0  7  4  2  2
  0  0  0  0  6  8  4  2  2
  0  0  0  0  4 12  6  4  2  2
  0  0  0  0  1 15 12  6  4  2  2
  0  0  0  0  0 17 15 10  6  4  2  2
  0  0  0  0  0 14 23 16 10  6  4  2  2
  0  0  0  0  0 10 30 23 14 10  6  4  2  2
  0  0  0  0  0  5 39 29 24 14 10  6  4  2  2
  0  0  0  0  0  1 42 42 31 22 14 10  6  4  2  2
Row 9 counts the following partitions:
  (432)   (54)     (63)      (72)       (81)        (9)
  (3321)  (333)    (621)     (711)      (21111111)  (111111111)
  (4221)  (441)    (6111)    (2211111)
  (4311)  (522)    (222111)  (3111111)
          (531)    (321111)
          (3222)   (411111)
          (5211)
          (22221)
          (32211)
          (33111)
          (42111)
          (51111)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
• Bridget Eileen Tenner, Reduced word manipulation: patterns and enumeration, J. Algebr. Comb. 46, No. 1, 189-217 (2017), table 1.
• Tewodros Amdeberhan, George E. Andrews, and Cristina Ballantine, Hook length and symplectic content in partitions, arXiv:2205.07322 [math.CO], 2022.
• Eric Weisstein's World of Mathematics, Graph Distance

Crossrefs

Row sums are A000041. Column sums are A071724.

Cf. A065770, A096771, A115720, A115994, A139582, A325169, A325183, A325188, A325195, A325200, A366157.

Programs

Mathematica

otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otbmax[#]==k&]],{n,0,15},{k,0,n}]

PARI

row(n)={my(r=vector(n+1)); forpart(p=n, my(w=0); for(i=1, #p, w=max(w,#p-i+p[i])); r[w+1]++); r} \\ Andrew Howroyd, Jan 12 2024

Formula

Sum_{k=1..n} k*T(n,k) = A366157(n). - Andrew Howroyd, Jan 12 2024