A027293 Triangular array given by rows: P(n,k) is the number of partitions of n that contain k as a part.

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 3, 2, 1, 1, 7, 5, 3, 2, 1, 1, 11, 7, 5, 3, 2, 1, 1, 15, 11, 7, 5, 3, 2, 1, 1, 22, 15, 11, 7, 5, 3, 2, 1, 1, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 77

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

Triangle read by rows in which row n lists the first n partition numbers A000041 in decreasing order. - Omar E. Pol, Aug 06 2011

A027293 * an infinite lower triangular matrix with A010815 (1, -1, -1, 0, 0, 1, ...) as the main diagonal the rest zeros = triangle A145975 having row sums = [1, 0, 0, 0, ...]. These matrix operations are equivalent to the comment in A010815 stating "when convolved with the partition numbers = [1, 0, 0, 0, ...]. - Gary W. Adamson, Oct 25 2008

From Gary W. Adamson, Oct 26 2008: (Start)

Row sums = A000070: (1, 2, 4, 7, 12, 19, 30, 45, 67, ...);

(this triangle)^2 = triangle A146023. (End)

(1) It appears that P(n,k) is also the total number of occurrences of k in the last k sections of the set of partitions of n (cf. A182703). (2) It appears that P(n,k) is also the difference, between n and n-k, of the total number of occurrences of k in all their partitions (cf. A066633). - Omar E. Pol, Feb 07 2012

Sequence B is called a reverse reluctant sequence of sequence A, if B is a triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. The present sequence is the reverse reluctant sequence of (A000041(k-1)){k>=0}. - _Boris Putievskiy, Dec 14 2012

Examples

			The triangle P begins (with offsets 0 it is Pa):
n \ k  1  2  3  4  5  6  7  8  9 10 ...
1:     1
2:     1  1
3:     2  1  1
4:     3  2  1  1
5:     5  3  2  1  1
6:     7  5  3  2  1  1
7:    11  7  5  3  2  1  1
8:    15 11  7  5  3  2  1  1
9:    22 15 11  7  5  3  2  1  1
10:   30 22 15 11  7  5  3  2  1  1
... reformatted by _Wolfdieter Lang_, Apr 14 2021
		

Links

• Robert Price, Table of n, a(n) for n = 1..5050
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Every column of P is A000041.

Cf. A145975, A010815.

Cf. A000070, A146023.

Cf. A182700, A027293, A067687, A155002.

Cf. A343234 (L-eigen-matrix).

Programs

Mathematica

f[n_] := Block[{t = Flatten[Union /@ IntegerPartitions@n]}, Table[Count[t, i], {i, n}]]; Array[f, 13] // Flatten
t[n_, k_] := PartitionsP[n-k]; Table[t[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 24 2014 *)

Formula

P(n,k) = p(n-k) = A000041(n-k), n>=1, k>=1. - Omar E. Pol, Feb 15 2013

a(n) = A000041(m), where m = (t*t + 3*t + 4)/2 - n, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 14 2012

From Wolfdieter Lang, Apr 14 2021: (Start)

Pa(n, m) = P(n+1, m+1) = A000041(n-m), for n >= m >= 0, and 0 otherwise, gives the Riordan matrix Pa = (P(x), x), of Toeplitz type, with the o.g.f. P(x) of A000041. The o.g.f. of triangle Pa (the o.g.f. of the row polynomials RPa(n, x) = Sum_{m=0..n} Pa(n, m)*x^m) is G(z, x) = P(z)/(1 - x*z).

The (infinite) matrix Pa has the 'L-eigen-sequence' B = A067687, that is, Pa*vec(B) = L*vec(B), with the matrix L with elements L(i, j) = delta(i, j-1) (Kronecker's delta symbol). For such L-eigen-sequences see the Bernstein and Sloane links under A155002.

Thanks to Gary W. Adamson for motivating me to look at such matrices and sequences. (End)