A325188 - OEIS

%I #14 Jan 12 2024 17:26:19

%S 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 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 0,2,17,32,5,0,0,0,0,0,0,0

%N 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 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 Andrew Howroyd, <a href="/A325188/b325188.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%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) = A368986(n).

%e Triangle begins:

%e   1

%e   0  1

%e   0  2  0

%e   0  2  1  0

%e   0  2  3  0  0

%e   0  2  5  0  0  0

%e   0  2  8  1  0  0  0

%e   0  2  9  4  0  0  0  0

%e   0  2 12  8  0  0  0  0  0

%e   0  2 13 15  0  0  0  0  0  0

%e   0  2 16 23  1  0  0  0  0  0  0

%e   0  2 17 32  5  0  0  0  0  0  0  0

%e   0  2 20 43 12  0  0  0  0  0  0  0  0

%e   0  2 21 54 24  0  0  0  0  0  0  0  0  0

%e   0  2 24 67 42  0  0  0  0  0  0  0  0  0  0

%e   0  2 25 82 66  1  0  0  0  0  0  0  0  0  0  0

%t otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];

%t Table[Length[Select[IntegerPartitions[n],otb[#]==k&]],{n,0,15},{k,0,n}]

%o (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 Row sums are A000041. Column k = 1 is A130130. Column k = 2 is A325168.

%Y Cf. A000245, A065770, A096771, A115994, A325169, A325183, A325187, A325189, A325191, A325195, A325200, A368986.

%K nonn,tabl

%O 0,5

%A _Gus Wiseman_, Apr 11 2019