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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.

%I A325188 #27 Aug 04 2025 11:20:45
%S A325188 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,
%T A325188 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,
%U A325188 0,2,17,32,5,0,0,0,0,0,0,0
%N 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.
%C A325188 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.
%H A325188 Andrew Howroyd, <a href="/A325188/b325188.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A325188 N. Guru Sharan, <a href="https://arxiv.org/abs/2507.20260">Rook decomposition of the Partition function</a>, arXiv:2507.20260 [math.CO], 2025. See p. 4.
%H A325188 N. Guru Sharan and Armin Straub, <a href="https://arxiv.org/abs/2507.19047">Partitions with Durfee triangles of fixed size</a>, arXiv:2507.19047 [math.CO], 2025. See p. 10.
%H A325188 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>.
%F A325188 Sum_{k=1..n} k*T(n,k) = A368986(n).
%e A325188 Triangle begins:
%e A325188   1
%e A325188   0  1
%e A325188   0  2  0
%e A325188   0  2  1  0
%e A325188   0  2  3  0  0
%e A325188   0  2  5  0  0  0
%e A325188   0  2  8  1  0  0  0
%e A325188   0  2  9  4  0  0  0  0
%e A325188   0  2 12  8  0  0  0  0  0
%e A325188   0  2 13 15  0  0  0  0  0  0
%e A325188   0  2 16 23  1  0  0  0  0  0  0
%e A325188   0  2 17 32  5  0  0  0  0  0  0  0
%e A325188   0  2 20 43 12  0  0  0  0  0  0  0  0
%e A325188   0  2 21 54 24  0  0  0  0  0  0  0  0  0
%e A325188   0  2 24 67 42  0  0  0  0  0  0  0  0  0  0
%e A325188   0  2 25 82 66  1  0  0  0  0  0  0  0  0  0  0
%t A325188 otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
%t A325188 Table[Length[Select[IntegerPartitions[n],otb[#]==k&]],{n,0,15},{k,0,n}]
%o A325188 (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
%Y A325188 Row sums are A000041.
%Y A325188 Columns k=1-4 give: A130130, A325168, A382682, A384562.
%Y A325188 Cf. A000245, A065770, A096771, A115994, A325169, A325183, A325187, A325189, A325191, A325195, A325200, A368986.
%K A325188 nonn,tabl
%O A325188 0,5
%A A325188 _Gus Wiseman_, Apr 11 2019