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A325200
Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is k.
13
1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 3, 0, 2, 0, 0, 3, 2, 0, 2, 0, 1, 0, 6, 2, 0, 2, 0, 0, 4, 3, 4, 2, 0, 2, 0, 0, 6, 2, 6, 4, 2, 0, 2, 0, 0, 4, 9, 5, 4, 4, 2, 0, 2, 0, 1, 0, 15, 6, 8, 4, 4, 2, 0, 2, 0, 0, 5, 12, 12, 9, 6, 4, 4, 2, 0, 2, 0, 0, 10, 6, 21, 10, 12, 6, 4, 4, 2, 0, 2, 0
OFFSET
0,5
FORMULA
Sum_{k=1..n} k*T(n,k) = A366157(n) - A368986(n). - Andrew Howroyd, Jan 13 2024
EXAMPLE
Triangle begins:
1
1 0
0 2 0
1 0 2 0
0 3 0 2 0
0 3 2 0 2 0
1 0 6 2 0 2 0
0 4 3 4 2 0 2 0
0 6 2 6 4 2 0 2 0
0 4 9 5 4 4 2 0 2 0
1 0 15 6 8 4 4 2 0 2 0
0 5 12 12 9 6 4 4 2 0 2 0
0 10 6 21 10 12 6 4 4 2 0 2 0
0 10 12 20 18 13 10 6 4 4 2 0 2 0
0 5 27 20 23 16 16 10 6 4 4 2 0 2 0
1 0 38 22 32 22 19 14 10 6 4 4 2 0 2 0
0 6 34 38 34 35 20 22 14 10 6 4 4 2 0 2 0
0 15 22 57 44 40 34 23 20 14 10 6 4 4 2 0 2 0
0 20 20 71 55 54 45 32 26 20 14 10 6 4 4 2 0 2 0
0 15 43 70 71 66 60 44 35 24 20 14 10 6 4 4 2 0 2 0
0 6 74 64 99 83 70 65 42 38 24 20 14 10 6 4 4 2 0 2 0
Row n = 9 counts the following partitions (empty columns not shown):
(432) (333) (54) (63) (72) (81) (9)
(3321) (441) (621) (6111) (711) (21111111) (111111111)
(4221) (522) (22221) (222111) (2211111)
(4311) (531) (51111) (411111) (3111111)
(3222) (321111)
(5211)
(32211)
(33111)
(42111)
MATHEMATICA
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&, Append[ptn, 0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&, Append[ptn, 0]];
Table[Length[Select[IntegerPartitions[n], otbmax[#]-otb[#]==k&]], {n, 0, 20}, {k, 0, n}]
PROG
(PARI) row(n)={my(r=vector(n+1)); forpart(p=n, my(b=#p, c=0); for(i=1, #p, my(x=#p-i+p[i]); b=min(b, x); c=max(c, x)); r[c-b+1]++); r} \\ Andrew Howroyd, Jan 12 2024
CROSSREFS
Row sums are A000041. Column k = 1 is A325191. Column k = 2 is A325199.
T(n,k) = A325189(n,k) - A325188(n,k).
Sequence in context: A352555 A307831 A217564 * A266909 A276491 A035177
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 11 2019
STATUS
approved