Symmetric Pascal matrices modulo \(p\). (English) Zbl 1055.15028
The authors present results and conjectures concerning symmetric matrices associated to Pascal’s triangle. Consider the symmetric matrix \(P(n)\) with coefficients \(p_{i,j}= \binom {i+j}{i}\), \(0\leq i,j< n\). We call \(P(n)\) the symmetric Pascal matrix of order \(n\). The entries of \(P(n)\) satisfy the recurrence: \(p_{i,j}= p_{i-1,j} + p_{i,j-1}\).
The authors first give a formula for the determinant over \(\mathbb{Z}\) of the reduction modulo 2 with values in \(\{0,1\}\) and of the reduction modulo 3 with values in \(\{-1,0,1\}\) for such a matrix. Then they study the reduction modulo a prime \(p\) of the characteristic polynomials of these matrices. The main results imply a recursive formula for the prime \(p= 2\) and a conjectural recursive formula for \(p= 3\).
The authors first give a formula for the determinant over \(\mathbb{Z}\) of the reduction modulo 2 with values in \(\{0,1\}\) and of the reduction modulo 3 with values in \(\{-1,0,1\}\) for such a matrix. Then they study the reduction modulo a prime \(p\) of the characteristic polynomials of these matrices. The main results imply a recursive formula for the prime \(p= 2\) and a conjectural recursive formula for \(p= 3\).
Reviewer: Rodica Covaci (Cluj-Napoca)
MSC:
| 15B36 | Matrices of integers |
| 15B57 | Hermitian, skew-Hermitian, and related matrices |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 15A21 | Canonical forms, reductions, classification |
| 11C20 | Matrices, determinants in number theory |
Keywords:
symmetric Pascal matrix; determinant; reduction; characteristic polynomials; recursive formulaOnline Encyclopedia of Integer Sequences:
Multiplicity of the factor (1+x+x^2) in the characteristic polynomial modulo 2 of the symmetric matrix with entries binomial(i+j,i), 0<=i,j<n.Multiplicity of the root 1 in the characteristic polynomial mod 2 of the n X n matrix with entries binomial(i+j,i), 0<=i,j<n.
a(n) = floor(S2(n)/2) mod 2, where S2(n) is the binary weight of n.
References:
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| [2] | Bacher, R., Determinants of matrices related to the Pascal triangle, J. Théor. des Nombres Bordeaux, 14, 19-41 (2002) · Zbl 1023.11011 |
| [3] | Krattenthaler, C., Advanced determinant calculus, Sém. Lothar. Combin. 42, B42q, 67 (1999) · Zbl 0923.05007 |
| [4] | Lunnon, W. F., The Pascal matrix, Fibonacci Quart., 15, 201-204 (1977) · Zbl 0376.15009 |
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