Heisenberg characters, unitriangular groups, and Fibonacci numbers

Let U n ( F q ) denote the group of unipotent n × n upper triangular matrices over a finite field with q elements. We show that the Heisenberg characters of U n + 1 ( F q ) are indexed by lattice paths from the origin to the line x + y = n using the steps ( 1 , 0 ) , ( 1 , 1 ) , ( 0 , 1 ) , ( 0 , 2 ) , which are labeled in a certain way by nonzero elements of F q . In particular, we prove for n ⩾ 1 that the number of Heisenberg characters of U n + 1 ( F q ) is a polynomial in q − 1 with nonnegative integer coefficients and degree n, whose leading coefficient is the nth Fibonacci number. Similarly, we find that the number of Heisenberg supercharacters of U n ( F q ) is a polynomial in q − 1 whose coefficients are Delannoy numbers and whose values give a q-analogue for the Pell numbers. By counting the fixed points of the action of a certain group of linear characters, we prove that the numbers of supercharacters, irreducible supercharacters, Heisenberg supercharacters, and Heisenberg characters of the subgroup of U n ( F q ) consisting of matrices whose superdiagonal entries sum to zero are likewise all polynomials in q − 1 with nonnegative integer coefficients.