|
|
A325165
|
|
Regular triangle read by rows where T(n,k) is the number of integer partitions of n whose inner lining partition has last (smallest) part equal to k.
|
|
8
|
|
|
1, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 4, 0, 2, 0, 0, 0, 5, 0, 3, 2, 0, 0, 0, 6, 0, 4, 4, 0, 0, 0, 0, 7, 0, 5, 6, 3, 0, 0, 0, 0, 8, 0, 7, 8, 6, 0, 0, 0, 0, 0, 9, 0, 9, 10, 9, 4, 0, 0, 0, 0, 0, 10, 0, 13, 12, 12, 8, 0, 0, 0, 0, 0, 0, 11
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,6
|
|
COMMENTS
|
The k-th part of the inner lining partition of an integer partition is the number of squares in its Young diagram that are k diagonal steps from the lower-right boundary. For example, the partition (6,5,5,3) has diagram
o o o o o o
o o o o o
o o o o o
o o o
which has diagonal distances from the lower-right boundary equal to
3 3 3 2 1 1
3 2 2 2 1
2 2 1 1 1
1 1 1
so the inner lining sequence is (9,6,4) with last part 4, so (6,5,5,3) is counted under T(19,4).
|
|
LINKS
|
|
|
FORMULA
|
G.f.: A(x,y) = 1 + Sum_{k>=1} x^(k^2)/((1 - y*x^k) * Product_{j=1..k-1} (1 - x^j))^2. - Andrew Howroyd, Jan 19 2023
|
|
EXAMPLE
|
Triangle begins:
1
0 1
0 0 2
0 0 0 3
0 1 0 0 4
0 2 0 0 0 5
0 3 2 0 0 0 6
0 4 4 0 0 0 0 7
0 5 6 3 0 0 0 0 8
0 7 8 6 0 0 0 0 0 9
0 9 10 9 4 0 0 0 0 0 10
0 13 12 12 8 0 0 0 0 0 0 11
0 17 16 15 12 5 0 0 0 0 0 0 12
0 24 20 18 16 10 0 0 0 0 0 0 0 13
0 31 28 21 20 15 6 0 0 0 0 0 0 0 14
0 42 36 27 24 20 12 0 0 0 0 0 0 0 0 15
0 54 50 33 28 25 18 7 0 0 0 0 0 0 0 0 16
0 71 64 45 32 30 24 14 0 0 0 0 0 0 0 0 0 17
0 90 86 57 40 35 30 21 8 0 0 0 0 0 0 0 0 0 18
Row n = 9 counts the following partitions (empty columns not shown):
(72) (63) (54) (9)
(333) (522) (432) (81)
(621) (531) (441) (711)
(5211) (4221) (3222) (6111)
(42111) (4311) (3321) (51111)
(321111) (32211) (22221) (411111)
(2211111) (33111) (3111111)
(222111) (21111111)
(111111111)
|
|
MATHEMATICA
|
pml[ptn_]:=If[ptn=={}, {}, FixedPointList[If[#=={}, {}, DeleteCases[Rest[#]-1, 0]]&, ptn][[-3]]];
Table[Length[Select[IntegerPartitions[n], Total[pml[#]]==k&]], {n, 0, 10}, {k, 0, n}]
|
|
PROG
|
(PARI) T(n) = {my(v=Vec(1+sum(k=1, sqrtint(n), x^(k^2)/((1-y*x^k)*prod(j=1, k-1, 1 - x^j + O(x^(n+1-k^2))))^2))); vector(#v, i, Vecrev(v[i], -i))}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 19 2023
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|