Shedding light on words

Abstract

A word over an alphabet [k] can be represented by a bargraph, where the height of the i-th column is the size of the i-th part. If North is in the direction of the positive y-axis and East is in the direction of the positive x-axis, a light source projects parallel rays from the North-West direction, at an angle of 45 degrees to the y-axis. These rays strike the cells of the bargraph. We say a cell is lit if the rays strike its West facing edge or North facing edge or both. With the use of matrix algebra we find the generating function that counts the number of lit cells. From this we find the average number of lit cells in a word of length n.

Counting water cells in bargraphs of compositions and set partitions

“…To determine the total number of water cells, suppose π = 1π (1) • • • kπ (k) ∈ P n,k , where each π ( ) is -ary for 1 ≤ ≤ k. We first count cells lying above a column corresponding to a letter r in π (j) for some 2 ≤ j ≤ k −1. Then π (j) = αrβ, where α and β are j-ary and possibly empty.…”

“…We treat the case when k is even, the odd case being similar which we leave to the reader. Let P * n,k denote the subset of P n,k consisting of π = 1π (1) 2π (2) • • • kπ (k) satisfying the following conditions: (i) π (2) = π (4)…”

“…To complete the proof of (11), it suffices to identify a wc-parity reversing involution of P n,k − P * n,k . Suppose π = 1π (1) • • • kπ (k) ∈ P n,k − P * n,k . Let i o be the largest index i ∈ [k − 1] (if it exists) such that either (a) i is odd and π (i) contains some letter s > 1, or (b) i is even with π (i) = ∅.…”

“…We first count all π containing one unit, i.e., r = 1. Then within π = 1 a1 π (1) 1 a2 the section π (1) contributes g i−1 (x, q) towards the gf, while the 1 a1 and 1 a2 sequences contribute x 1−x and (Note that each 1 within the 1 a2 sequence contributes xq i−1 due to the understood letter i that follows π.) This accounts for the first term on the right side of (12).…”

“…Prellberg and Brak [23] and Feretíc [12] were among the first to study statistics on bargraphs and found a generating function with two variables x and y keeping track of the number of horizontal and up steps, respectively. Blecher et al later considered levels [3], peaks [4], descents [5], area [5] and other statistics [1,6], and further refined counts were given by Deutsch and Elizalde in [10], where bargraphs were viewed as Motzkin paths without peaks or valleys. There is also some interest in generating function formulas related to bargraphs in statistical physics [21,22], where they are used to model certain kinds of polymers.…”