A385901 - OEIS

A385901

Number of figurative partitions of n (weakly decreasing index paths).

1, 1, 2, 3, 4, 5, 6, 7, 10, 11, 13, 17, 19, 22, 28, 33, 38, 46, 53, 62, 73, 83, 96, 116, 131, 151, 178, 203, 231, 269, 302, 345, 399, 450, 510, 589, 661, 749, 858, 963, 1082, 1231, 1374, 1547, 1752, 1952, 2188, 2469, 2751, 3073, 3454, 3842, 4279, 4798, 5318, 5909

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

We study weakly decreasing index paths in two-dimensional arrays that sum up to a given total t if evaluated by a generating function A. The arrays are denoted by A(n, k) for n >= 0 and k >= 0. A path is a list of index pairs W = [i_0, i_1, ..., i_m] where the index pairs i_j = (n_j, k_j) are subject to the condition that n_j > n_{j+1} and k_j >= k_{j+1}, and the values of the generating function A sum to a prescribed positive number t = Sum_{(n, k) in W} A(n, k).

Here we consider the array A139600 with the generating function P(n, k) = k + n * (k - 1) * k / 2 for n >= 0, k >= 2, supplemented by the condition P(n, 1) = 1 if n = 0 otherwise 0.

We call F = [(i, j) in W: P(i, j)] a 'figurative partition of n with weakly decreasing indices' if Sum(F) = n and call n the 'shape' and k the 'size' of P(n, k).

LINKS

Table of n, a(n) for n=1..56.

Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.

Peter Luschny, List of the partitions for 1 <= n <= 12.

Index to sequences related to polygonal numbers.

EXAMPLE

Some figurative partitions of 27 are, in the format '(shape, size) value':

(6, 3) 21 + (2, 2) 4 + (0, 2) 2;

(5, 3) 18 + (3, 2) 5 + (1, 2) 3 + (0, 1) 1;

(5, 3) 18 + (1, 3) 6 + (0, 3) 3;

(4, 3) 15 + (3, 2) 5 + (2, 2) 4 + (1, 2) 3.

CROSSREFS