### Short witnesses for Parikh-friendly permutations

Jamie Simpson
2020 The Australasian Journal of Combinatorics
showed that for every permutation π of the ordered alphabet A = {a 1 , a 2 , . . . , a n } there exists a word w ∈ A * , in which each letter of A appears at least once, such that w and π(w) have the same Parikh matrix. He conjectured that it is always possible to find such a w whose length is at most 2n. We prove this. We use the usual notation for combinatorics on words. A word of m elements is x = x[1 . . m], with x[i] being the ith element and x[i . . j] the factor of elements from position
more » ... i to position j. If i = 1 then the factor is a prefix and if j = m then it is a suffix. The reverse of x, written x, is the word [1] . If a word equals its own reverse then it is a palindrome. The letters in x come from some alphabet. We will use the ordered alphabet A = {a 1 , a 2 , . . . , a n } where a 1 < a 2 < • • • < a n , except in some examples where we use the more familiar a < b < c < • • • . The set of all finite words with letters from A is A * . The length of x, written |x|, is the number of letters in x. If i ≤ j we use a i,j as an abbreviation for a i a i+1 . . . a j with a i,i interpreted as The number of occurrences of u as a subword of w is written |w| u . For example the word w = cbabbacb contains 4 occurrences of the subword ab so |w| ab = 4. The number of occurrences of the single letter c in w is |w| c = 2. Note that ca is a subword of w but not a factor. The set of letters occurring in a word w is written alph(w). This note concerns Parikh Matrices which may be defined as follows (for an alternative but equivalent definition see [3] ). The Parikh matrix for a word w defined on the alphabet {a 1 , a 2 , . . . , a n } is an (n + 1) × (n + 1) matrix Ψ(w) with entries ψ i,j where ψ i,j = 0 for 1 ≤ j < i ≤ n + 1, ψ i,i = 1 for 1 ≤ i ≤ n + 1, ψ i,j+1 = |w| a i,j for 1 ≤ i ≤ j ≤ n.