A385900 - OEIS

A385900

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

1, 1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 4, 6, 8, 7, 7, 9, 7, 7, 9, 9, 10, 11, 10, 9, 13, 12, 10, 13, 13, 13, 13, 13, 15, 16, 16, 18, 20, 17, 15, 17, 17, 18, 21, 21, 21, 22, 20, 23, 26, 26, 25, 30, 25, 25, 28, 29, 27, 32, 29, 25, 32, 29, 29, 37, 39, 35, 42, 44, 38

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

OFFSET

1,3

COMMENTS

We study strictly 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). We call n the 'color' and k the 'shade' of 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.

The array starts with the first row being the nonnegative integers, then triangular numbers, squares, pentagons, etc., the columns begin at k = 2, and the case k = 1 is treated as an exceptional case. We call F = [(i, j) in W: P(i, j)] a 'figurative partition of n' if Sum(F) = n. In the context of figurative partitions, we call n the 'shape' and k the 'size' of P(n, k).

For example, the path W = [(3, 4), (2, 2), (0, 1)] leads to F = 22 + 4 + 1, which is a figurative partition of 27. In other words, 27 has a figurative layout on a counting board with a pentagon of size 4, a square of size 2, and a 'pebble', the unique figurative partition of 1. Another figurative partition of 27 is an octagon of size 3, a pentagon of size 2, plus a pebble.

The question is: given a positive integer n, how many figurative partitions of n exist?

LINKS

Table of n, a(n) for n=1..72.

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

The 15 figurative partitions of 42, in the format '(shape, size) value', are:

(40, 2) 42;

(39, 2) 41 + (0, 1) 1;

(13, 3) 42;

(12, 3) 39 + (1, 2) 3;