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.

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

Offset: 0

Author

Gus Wiseman, Apr 05 2019

Keywords

• nonn
• tabl

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).

Examples

			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)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

Row sums are A000041. Column k = 1 is A188674.

Cf. A006918, A115720, A115994, A117485, A252464, A257990, A325163, A325166, A325168.

Programs

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}]

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

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