id: 01153461
dt: j
an: 01153461
au: McDougal, K.
ti: Locally invariant positions of (0, 1) matrices.
so: Ars Comb. 44, 219-224 (1996).
py: 1996
pu: Charles Babbage Research Centre, Winnipeg, MB
la: EN
cc: 
ut: $(0,1)$-matrix; invariant position; locally invariant position
ci: Zbl 0112.24806; Zbl 0448.05012
li: 
ab: Given nonnegative integral vectors $R=(r_1,r_2,\ldots ,r_m)$ and
    $S=(s_1,s_2,\ldots ,s_n)$, the class of all $(0,1)$ matrices with row
    sum vector $R$ and column sum vector $S$ is denoted by $A(R,S)$. A
    position $ij$ is called invariant for $A(R,S)$ if all matrices from
    $A(R,S)$ have the same entry on this position (either all have 1 or all
    0). {\it H. J. Ryser} [Combinatorial mathematics, Carus Mathematical
    Monograph No. 14, Math. Association of America (1963; Zbl 0112.24806)]
    showed that any matrix of $A(R,S)$ can be transformed to any other
    matrix of $A(R,S)$ by a series of interchanges (replacing of a
    submatrix $\binom {1 0}{0 1}$ by a submatrix $\binom {0 1}{1 0}$). In
    an attempt to generalize the concept of invariant position studied in
    {\it R. A. Brualdi} and {\it J. A. Ross} [Proc. Am. Math. Soc. 80,
    706-710 (1980; Zbl 0448.05012)], the author introduces and investigates
    the concept of locally invariant position (position $ij$ is locally
    invariant for $A(R,S)$ iff there exists a matrix $A\in A(R,S)$ such
    that no single interchange applied to $A$ changes the value of the
    $ij$-entry). One of the main results states that if $ij$ is a locally
    invariant 1-position then so is every position $vw$ for $1\le v\le i$,
    $1\le w\le j$.
rv: Jan Kratochvíl (Praha)