Introduction Distribution of 3–factorizations S + and S − sums Denumerants from S ± sums Some comments on denumerant of numerical 3–semigroups IMNS 2014 , Cortona Francesc Aguiló-Gost, Dept. MA-IV, Univ. Politècnica de Catalunya, Barcelona Pedro A. García-Sánchez, Depto. de Álgebra, Univ. de Granada, Granada David Llena, Depto. de Matemáticas, Univ. de Almería, Almería Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Definitions and notation S + and S − sums Known results Denumerants from S ± sums Introduction Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Definitions and notation S + and S − sums Known results Denumerants from S ± sums Definitions and notation. Given a 1 , . . . , a n ∈ N , with 1 ≤ a 1 < a 2 < ... < a n and gcd( a 1 , ..., a n ) = 1 , the numerical n –semigrup S = � a 1 , . . . , a n � is defined by � a 1 , . . . , a n � = { x 1 a 1 + · · · + x n a n : ( x 1 , ..., x n ) ∈ N n } . Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Definitions and notation S + and S − sums Known results Denumerants from S ± sums Definitions and notation. Given a 1 , . . . , a n ∈ N , with 1 ≤ a 1 < a 2 < ... < a n and gcd( a 1 , ..., a n ) = 1 , the numerical n –semigrup S = � a 1 , . . . , a n � is defined by � a 1 , . . . , a n � = { x 1 a 1 + · · · + x n a n : ( x 1 , ..., x n ) ∈ N n } . Given m ∈ S , a vector ( x 1 , ..., x n ) ∈ N n is a factorization of m if x 1 a 1 + · · · + x n a n = m . Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Definitions and notation S + and S − sums Known results Denumerants from S ± sums Definitions and notation. Given a 1 , . . . , a n ∈ N , with 1 ≤ a 1 < a 2 < ... < a n and gcd( a 1 , ..., a n ) = 1 , the numerical n –semigrup S = � a 1 , . . . , a n � is defined by � a 1 , . . . , a n � = { x 1 a 1 + · · · + x n a n : ( x 1 , ..., x n ) ∈ N n } . Given m ∈ S , a vector ( x 1 , ..., x n ) ∈ N n is a factorization of m if x 1 a 1 + · · · + x n a n = m . { ( x 1 , ..., x n ) ∈ N n : x 1 a 1 + · · · + x n a n = m } , F ( m, � a 1 , ..., a n � ) = Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Definitions and notation S + and S − sums Known results Denumerants from S ± sums Definitions and notation. Given a 1 , . . . , a n ∈ N , with 1 ≤ a 1 < a 2 < ... < a n and gcd( a 1 , ..., a n ) = 1 , the numerical n –semigrup S = � a 1 , . . . , a n � is defined by � a 1 , . . . , a n � = { x 1 a 1 + · · · + x n a n : ( x 1 , ..., x n ) ∈ N n } . Given m ∈ S , a vector ( x 1 , ..., x n ) ∈ N n is a factorization of m if x 1 a 1 + · · · + x n a n = m . { ( x 1 , ..., x n ) ∈ N n : x 1 a 1 + · · · + x n a n = m } , F ( m, � a 1 , ..., a n � ) = d( m, � a 1 , ..., a n � ) = |F ( m, � a 1 , ..., a n � ) | . Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Definitions and notation S + and S − sums Known results Denumerants from S ± sums Definitions and notation Given N ∈ S = � a 1 , ..., a n � , the Apéry set Ap( S, N ) is Ap( S, N ) = { s ∈ S : s − N �∈ S } . Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Definitions and notation S + and S − sums Known results Denumerants from S ± sums Know results Generating function of d( m, � a 1 , ..., a n � ) (Sylvester, 1882): 1 φ ( z ) = (1 − z a 1 )(1 − z a 2 ) · · · (1 − z a n ) . Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Definitions and notation S + and S − sums Known results Denumerants from S ± sums Know results Generating function of d( m, � a 1 , ..., a n � ) (Sylvester, 1882): 1 φ ( z ) = (1 − z a 1 )(1 − z a 2 ) · · · (1 − z a n ) . Assymptotic behaviour (Schur, 1926): d( m, � a 1 , ..., a n � ) lim sup = 1 . m n − 1 m →∞ a 1 a 2 ··· a n ( n − 1)! Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Definitions and notation S + and S − sums Known results Denumerants from S ± sums Know results Denumerant formula for n = 2 (Popoviciu, 1953) d( m, � p, q � ) = m + pf ( m ) + qg ( m ) − 1 lcm( p, q ) where f ( m ) ≡ − mp − 1 (mod q ) and g ( m ) ≡ − mq − 1 (mod p ) . Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Definitions and notation S + and S − sums Known results Denumerants from S ± sums Know results Denumerant formula for n = 2 (Popoviciu, 1953) d( m, � p, q � ) = m + pf ( m ) + qg ( m ) − 1 lcm( p, q ) where f ( m ) ≡ − mp − 1 (mod q ) and g ( m ) ≡ − mq − 1 (mod p ) . Recursive denumerant formulae for n = 3 , 4 (Ehrhart, 1967). Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Definitions and notation S + and S − sums Known results Denumerants from S ± sums Know results Denumerant formula for n = 2 (Popoviciu, 1953) d( m, � p, q � ) = m + pf ( m ) + qg ( m ) − 1 lcm( p, q ) where f ( m ) ≡ − mp − 1 (mod q ) and g ( m ) ≡ − mq − 1 (mod p ) . Recursive denumerant formulae for n = 3 , 4 (Ehrhart, 1967). For n = 3 , an O ( ab ) algorithm for computing d( m, � a, b, c � ) was given by Lisoněk 1995. Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Plane distribution S + and S − sums Denumerant sums Denumerants from S ± sums Distribution of 3–factorizations Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
35 40 45 50 55 60 65 70 75 80 28 33 38 43 48 53 58 63 68 73 21 26 31 36 41 46 51 56 61 66 14 19 24 29 34 39 44 49 54 59 7 12 17 22 27 32 37 42 47 52 0 5 10 15 20 25 30 35 40 45 Introduction Distribution of 3–factorizations Plane distribution S + and S − sums Denumerant sums Denumerants from S ± sums Plane distribution Every numerical 3–semigroup S = � a, b, c � has related a plane L-shape H that encodes the set Ap( S, c ) . Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
35 40 45 50 55 60 65 70 75 80 28 33 38 43 48 53 58 63 68 73 21 26 31 36 41 46 51 56 61 66 14 19 24 29 34 39 44 49 54 59 7 12 17 22 27 32 37 42 47 52 0 5 10 15 20 25 30 35 40 45 Introduction Distribution of 3–factorizations Plane distribution S + and S − sums Denumerant sums Denumerants from S ± sums Plane distribution Every numerical 3–semigroup S = � a, b, c � has related a plane L-shape H that encodes the set Ap( S, c ) . An L-shape is denoted by the lengths of her sides H = L( l, h, w, y ) . Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
35 40 45 50 55 60 65 70 75 80 28 33 38 43 48 53 58 63 68 73 21 26 31 36 41 46 51 56 61 66 14 19 24 29 34 39 44 49 54 59 7 12 17 22 27 32 37 42 47 52 0 5 10 15 20 25 30 35 40 45 Introduction Distribution of 3–factorizations Plane distribution S + and S − sums Denumerant sums Denumerants from S ± sums Plane distribution Every numerical 3–semigroup S = � a, b, c � has related a plane L-shape H that encodes the set Ap( S, c ) . An L-shape is denoted by the lengths of her sides H = L( l, h, w, y ) . Example: Ap( � 5 , 7 , 11 � , 11) = { 0 , 5 , 7 , 10 , 12 , 14 , 15 , 17 , 19 , 20 , 24 } Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Plane distribution S + and S − sums Denumerant sums Denumerants from S ± sums Plane distribution Every numerical 3–semigroup S = � a, b, c � has related a plane L-shape H that encodes the set Ap( S, c ) . An L-shape is denoted by the lengths of her sides H = L( l, h, w, y ) . Example: Ap( � 5 , 7 , 11 � , 11) = { 0 , 5 , 7 , 10 , 12 , 14 , 15 , 17 , 19 , 20 , 24 } 35 40 45 50 55 60 65 70 75 80 28 33 38 43 48 53 58 63 68 73 21 26 31 36 41 46 51 56 61 66 14 19 24 29 34 39 44 49 54 59 7 12 17 22 27 32 37 42 47 52 0 5 10 15 20 25 30 35 40 45 H = L(5 , 3 , 2 , 2) Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
Introduction Distribution of 3–factorizations Plane distribution S + and S − sums Denumerant sums Denumerants from S ± sums Plane distribution Every numerical 3–semigroup S = � a, b, c � has related a plane L-shape H that encodes the set Ap( S, c ) . An L-shape is denoted by the lengths of her sides H = L( l, h, w, y ) . Example: Ap( � 5 , 7 , 11 � , 11) = { 0 , 5 , 7 , 10 , 12 , 14 , 15 , 17 , 19 , 20 , 24 } 2 7 1 6 0 5 10 4 9 3 6 0 5 10 4 9 3 8 2 7 10 4 9 3 8 2 7 1 6 0 3 8 2 7 1 6 0 5 10 4 7 1 6 0 5 10 4 9 3 8 0 5 10 4 9 3 8 2 7 1 H = L(5 , 3 , 2 , 2) Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona
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