From Euler numbers to the golden ratio: rigidity, ranges and the most ambiguous Parikh matrices for words with repeated letters

The degree of ambiguity of a Parikh matrix, in Salomaa’s sense, is the number of words that share it. For permutations the adjacent pair counts determine the matrix, and the most ambiguous matrices are the two alternating ones, each shared by an Euler number of words (classical facts on descent sets, where the author’s 2006 thesis stopped). With repeated letters a fibre of the pair counts can split into several Parikh classes. For every content we characterise the pair-count vectors that determine the whole matrix: the condition is local, one of four arithmetic rules on every three consecutive letters. For three letters we find the least and greatest triple count given both pair counts; the least remainder in the Cauchy inequality of Mateescu, Salomaa and Yu is $\min(rs,(b-r)(b-s))$, where $r$ and $s$ are the residues of the pair counts modulo the middle multiplicity $b$. When the triple count is not forced, its two greatest values differ by two exactly on an arithmetic family and two corner families, and by one elsewhere. For content $(a,b,a)$ every largest pair-count fibre lies on the antidiagonal $x+y=ab$, and for $(2,b,2)$ with $b\ge16$ the largest Parikh class has $\lfloor(b^2+9b+36)/14\rfloor$ words. For $(a,2,c)$ with $a\le c$ the largest Parikh class has $\binom{a+c}{a}$ words if and only if $c(c-1)\le a(a+c-1)$, a threshold asymptotic to the golden ratio, with equality exactly at $(F_nF_{n+1},F_nF_{n+2})$, $n$ odd, a classical family; the largest pair-count fibre is the centre, uniquely unless $a=c=1$. For $k$ letters each occurring $n$ times, a total-positivity argument, with exact counts for $n\le3$, shows that the alternating classes are never the largest when $n\ge4$ and $k\ge3$, or $n=3$ and $k\ge4$, while for $n=2$ every class differing from them at one end loses; we conjecture that for $n=2$ they are the unique largest classes once $k\ge6$, as verified for $k\le12$.