A238875 Subdiagonal partitions: number of partitions (p1, p2, p3, ...) of n with pi <= i.

1, 1, 1, 2, 2, 4, 5, 7, 10, 15, 18, 26, 35, 47, 61, 80, 103, 138, 175, 224, 283, 362, 455, 577, 721, 898, 1111, 1380, 1701, 2106, 2577, 3156, 3844, 4680, 5671, 6879, 8312, 10034, 12060, 14478, 17319, 20715, 24703, 29442, 35004, 41578, 49247, 58278, 68796, 81132, 95502, 112320, 131877, 154705, 181158, 211908, 247475

Offset

0,4

Comments

The partitions are represented as weakly increasing lists of parts.

Partitions with subdiagonal growth (A238876) with first part = 1.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

G.f.: Sum_{n>=0} x^n * P(n) where P(n) is the row polynomial of the n-th row of A129176. This works because A129176(j,k) is also the number of subdiagonal partitions of j+k with j parts. - John Tyler Rascoe, Jan 20 2024

Example

The a(11) = 26 such partitions of 11 are:
01:  [ 1 1 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 1 1 1 3 ]
04:  [ 1 1 1 1 1 1 1 2 2 ]
05:  [ 1 1 1 1 1 1 1 4 ]
06:  [ 1 1 1 1 1 1 2 3 ]
07:  [ 1 1 1 1 1 1 5 ]
08:  [ 1 1 1 1 1 2 2 2 ]
09:  [ 1 1 1 1 1 2 4 ]
10:  [ 1 1 1 1 1 3 3 ]
11:  [ 1 1 1 1 1 6 ]
12:  [ 1 1 1 1 2 2 3 ]
13:  [ 1 1 1 1 2 5 ]
14:  [ 1 1 1 1 3 4 ]
15:  [ 1 1 1 2 2 2 2 ]
16:  [ 1 1 1 2 2 4 ]
17:  [ 1 1 1 2 3 3 ]
18:  [ 1 1 1 3 5 ]
19:  [ 1 1 1 4 4 ]
20:  [ 1 1 2 2 2 3 ]
21:  [ 1 1 2 2 5 ]
22:  [ 1 1 2 3 4 ]
23:  [ 1 1 3 3 3 ]
24:  [ 1 2 2 2 2 2 ]
25:  [ 1 2 2 2 4 ]
26:  [ 1 2 2 3 3 ]

Prog

(PARI) \\ here b: nr parts; k: max part, b+w-1: partition sum.
seq(n)={my(M=matrix(n, 1), v=vector(n+1)); M[1, 1]=v[1]=v[2]=1; for(b=2, n, M=matrix(n-b+1, b, w, k, if(w>=k, sum(j=1, min(b-1, k), M[w+1-k, j]))); v+=concat(vector(b), vecsum(Vec(M))~)); v} \\ Andrew Howroyd, Jan 19 2024
(PARI)
N=55;
VP=vector(N+1);  VP[1] =VP[2] = 1;  \\ one-based; memoization
P(n) = VP[n+1];
for (n=2, N, VP[n+1] = sum( i=0, n-1, P(i) * P(n-1 -i) * x^((i+1)*(n-1-i)) ) );
x='x+O('x^N);
A(x) = sum(n=0, N, x^n * P(n) );
Vec(A(x)) \\ Joerg Arndt, Jan 23 2024

Crossrefs

Cf. A008930 (subdiagonal compositions), A010054 (subdiagonal partitions into distinct parts).

Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).

Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).

Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).

Cf. A129176 and A227543.

Sequence in context: A350837 A239945 A363740 * A344740 A241391 A241736

Adjacent sequences: A238872 A238873 A238874 * A238876 A238877 A238878

Keyword

nonn

Author

Joerg Arndt, Mar 24 2014

Status

approved