# $(0,1)$-matrices, discrepancy and preservers

Czechoslovak Mathematical Journal (2019)

- Volume: 69, Issue: 4, page 1123-1131
- ISSN: 0011-4642

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topBeasley, LeRoy B.. "$(0,1)$-matrices, discrepancy and preservers." Czechoslovak Mathematical Journal 69.4 (2019): 1123-1131. <http://eudml.org/doc/294867>.

@article{Beasley2019,

abstract = {Let $m$ and $n$ be positive integers, and let $R = (r_1, \ldots , r_m)$ and $S = (s_1,\ldots , s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m \times n$$(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F(R)$ be the $m \times n$$(0,1)$-matrix, where for each $i$, the $i$th row of $F(R,S)$ consists of $r_i$ 1’s followed by $(n-r_i)$ 0’s. Let $A\in A(R,S)$. The discrepancy of A, $\{\rm disc\}(A)$, is the number of positions in which $F(R)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m\times n$ matrices over the binary Boolean semiring to itself that preserve sets related to the discrepancy. In particular, we show that bijective linear preservers of Ferrers matrices are either the identity mapping or, when $m=n$, the transpose mapping.},

author = {Beasley, LeRoy B.},

journal = {Czechoslovak Mathematical Journal},

keywords = {Ferrers matrix; row-dense matrix; discrepancy; linear preserver; strong linear preserver},

language = {eng},

number = {4},

pages = {1123-1131},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {$(0,1)$-matrices, discrepancy and preservers},

url = {http://eudml.org/doc/294867},

volume = {69},

year = {2019},

}

TY - JOUR

AU - Beasley, LeRoy B.

TI - $(0,1)$-matrices, discrepancy and preservers

JO - Czechoslovak Mathematical Journal

PY - 2019

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 69

IS - 4

SP - 1123

EP - 1131

AB - Let $m$ and $n$ be positive integers, and let $R = (r_1, \ldots , r_m)$ and $S = (s_1,\ldots , s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m \times n$$(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F(R)$ be the $m \times n$$(0,1)$-matrix, where for each $i$, the $i$th row of $F(R,S)$ consists of $r_i$ 1’s followed by $(n-r_i)$ 0’s. Let $A\in A(R,S)$. The discrepancy of A, ${\rm disc}(A)$, is the number of positions in which $F(R)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m\times n$ matrices over the binary Boolean semiring to itself that preserve sets related to the discrepancy. In particular, we show that bijective linear preservers of Ferrers matrices are either the identity mapping or, when $m=n$, the transpose mapping.

LA - eng

KW - Ferrers matrix; row-dense matrix; discrepancy; linear preserver; strong linear preserver

UR - http://eudml.org/doc/294867

ER -

## References

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- Brualdi, R. A., Shen, J., Discrepancy of matrices of zeros and ones, Electron. J. Comb. 6 (1999), Research Paper 15, 12 pages. (1999) Zbl0918.05029MR1674136
- Motlaghian, S. M., Armandnejad, A., Hall, F. J., 10.1007/s10587-016-0296-4, Czech. Math. J. 66 (2016), 847-858. (2016) Zbl06644037MR3556871DOI10.1007/s10587-016-0296-4

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