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%I #28 Jan 20 2024 04:10:44
%S 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,
%T 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,
%U 0,0,0,0,17,15,10,6,4,2,2
%N 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.
%C 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.
%H Andrew Howroyd, <a href="/A325189/b325189.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H Bridget Eileen Tenner, <a href="https://doi.org/10.1007/s10801-017-0752-8">Reduced word manipulation: patterns and enumeration</a>, J. Algebr. Comb. 46, No. 1, 189-217 (2017), table 1.
%H Tewodros Amdeberhan, George E. Andrews, and Cristina Ballantine, <a href="https://arxiv.org/abs/2205.07322">Hook length and symplectic content in partitions</a>, arXiv:2205.07322 [math.CO], 2022.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>
%F Sum_{k=1..n} k*T(n,k) = A366157(n). - _Andrew Howroyd_, Jan 12 2024
%e Triangle begins:
%e 1
%e 0 1
%e 0 0 2
%e 0 0 1 2
%e 0 0 0 3 2
%e 0 0 0 3 2 2
%e 0 0 0 1 6 2 2
%e 0 0 0 0 7 4 2 2
%e 0 0 0 0 6 8 4 2 2
%e 0 0 0 0 4 12 6 4 2 2
%e 0 0 0 0 1 15 12 6 4 2 2
%e 0 0 0 0 0 17 15 10 6 4 2 2
%e 0 0 0 0 0 14 23 16 10 6 4 2 2
%e 0 0 0 0 0 10 30 23 14 10 6 4 2 2
%e 0 0 0 0 0 5 39 29 24 14 10 6 4 2 2
%e 0 0 0 0 0 1 42 42 31 22 14 10 6 4 2 2
%e Row 9 counts the following partitions:
%e (432) (54) (63) (72) (81) (9)
%e (3321) (333) (621) (711) (21111111) (111111111)
%e (4221) (441) (6111) (2211111)
%e (4311) (522) (222111) (3111111)
%e (531) (321111)
%e (3222) (411111)
%e (5211)
%e (22221)
%e (32211)
%e (33111)
%e (42111)
%e (51111)
%t otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
%t Table[Length[Select[IntegerPartitions[n],otbmax[#]==k&]],{n,0,15},{k,0,n}]
%o (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
%Y Row sums are A000041. Column sums are A071724.
%Y Cf. A065770, A096771, A115720, A115994, A139582, A325169, A325183, A325188, A325195, A325200, A366157.
%K nonn,tabl
%O 0,6
%A _Gus Wiseman_, Apr 11 2019
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