OFFSET
0,5
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
FORMULA
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
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 11 2019
STATUS
approved