Total nonnegativity of some combinatorial matrices Abstract   Many combinatorial matrices xe2x80x94 such as those of binomial coefficients, Stirling numbers of both kinds, and Lah numbers xe2x80x94 are known to be totally nonnegative, meaning that all minors (determinants of square submatrices) are nonnegative.  The examples noted above can be placed in a common framework: for each one there is a nondecreasing sequence    (    a    1    ,    a    2    ,  xe2x80xa6  )   , and a sequence    (    e    1    ,    e    2    ,  xe2x80xa6  )   , such that the    (  m  ,  k  )    entry of the matrix is the coefficient of the polynomial    (  x  xe2x88x92    a    1    )  xe2x8bxaf  (  x  xe2x88x92    a    k    )    in the expansion of    (  x  xe2x88x92    e    1    )  xe2x8bxaf  (  x  xe2x88x92    e    m    )    as a linear combination of the polynomials    1  ,  x  xe2x88x92    a    1    ,  xe2x80xa6  ,  (  x  xe2x88x92    a    1    )  xe2x8bxaf  (  x  xe2x88x92    a    m    )   .  We consider this general framework. For a nondecreasing sequence    (    a    1    ,    a    2    ,  xe2x80xa6  )    we establish necessary and sufficient conditions on the sequence    (    e    1    ,    e    2    ,  xe2x80xa6  )    for the corresponding matrix to be totally nonnegative. As corollaries we obtain total nonnegativity of matrices of rook numbers of Ferrers boards, and of graph Stirling numbers of chordal graphs.

Conjectured Statistics for the Higher q;tCatalan Sequences This article describes conjectured combinatorial interpretations for the higher q,tCatalan sequences introduced by Garsia and Haiman, which arise in the theory of symmetric functions and Macdonald polynomials. We define new combinatorial statistics generalizing those proposed by Haglund  and Haiman for the original q,tCatalan sequence. We prove explicit summation formulas, bijections, and recursions involving the new statistics. We show that specializations of the combinatorial sequences obtained by setting t1 or q1 or t1q agree with the corresponding specializations of the GarsiaHaiman sequences. A third statistic occurs naturally in the combinatorial setting,  leading to the introduction of q,t,rCatalan sequences.  Similar combinatorial results are proved for these trivariate sequences.