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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.

%I A325189 #29 Feb 16 2025 08:33:58
%S A325189 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 A325189 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 A325189 0,0,0,0,17,15,10,6,4,2,2
%N 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.
%C A325189 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 A325189 Andrew Howroyd, <a href="/A325189/b325189.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A325189 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 A325189 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 A325189 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>
%F A325189 Sum_{k=1..n} k*T(n,k) = A366157(n). - _Andrew Howroyd_, Jan 12 2024
%e A325189 Triangle begins:
%e A325189   1
%e A325189   0  1
%e A325189   0  0  2
%e A325189   0  0  1  2
%e A325189   0  0  0  3  2
%e A325189   0  0  0  3  2  2
%e A325189   0  0  0  1  6  2  2
%e A325189   0  0  0  0  7  4  2  2
%e A325189   0  0  0  0  6  8  4  2  2
%e A325189   0  0  0  0  4 12  6  4  2  2
%e A325189   0  0  0  0  1 15 12  6  4  2  2
%e A325189   0  0  0  0  0 17 15 10  6  4  2  2
%e A325189   0  0  0  0  0 14 23 16 10  6  4  2  2
%e A325189   0  0  0  0  0 10 30 23 14 10  6  4  2  2
%e A325189   0  0  0  0  0  5 39 29 24 14 10  6  4  2  2
%e A325189   0  0  0  0  0  1 42 42 31 22 14 10  6  4  2  2
%e A325189 Row 9 counts the following partitions:
%e A325189   (432)   (54)     (63)      (72)       (81)        (9)
%e A325189   (3321)  (333)    (621)     (711)      (21111111)  (111111111)
%e A325189   (4221)  (441)    (6111)    (2211111)
%e A325189   (4311)  (522)    (222111)  (3111111)
%e A325189           (531)    (321111)
%e A325189           (3222)   (411111)
%e A325189           (5211)
%e A325189           (22221)
%e A325189           (32211)
%e A325189           (33111)
%e A325189           (42111)
%e A325189           (51111)
%t A325189 otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
%t A325189 Table[Length[Select[IntegerPartitions[n],otbmax[#]==k&]],{n,0,15},{k,0,n}]
%o A325189 (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 A325189 Row sums are A000041. Column sums are A071724.
%Y A325189 Cf. A065770, A096771, A115720, A115994, A139582, A325169, A325183, A325188, A325195, A325200, A366157.
%K A325189 nonn,tabl
%O A325189 0,6
%A A325189 _Gus Wiseman_, Apr 11 2019