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Partitions sans petites parts (II)

Élie Mosaki (2008)

Journal de Théorie des Nombres de Bordeaux

On désigne par r ( n , m ) le nombre de partitions de l’entier n en parts supérieures ou égales à m , et R ( n , m ) = r ( n - m , m ) le nombre de partitions de n de plus petite part m . Dans un précédent article (voir [9]) un développement asymptotique de r ( n , m ) est obtenu uniformément pour 1 ≤ m = O ( n )  ; on complète ce développement uniformément pour 1 ≤ m = ( n log - 3 n ) . Afin de prolonger les résultats jusqu’à m ≤ n , on donne un encadrement de r ( n , m ) valable pour n 2 / 3 ≤ m ≤ n en utilisant la relation r ( n , m ) = ∑ t = 1 ⌊ n / m ⌋ P ( n - ( m - 1 ) t , t ) où P ( i , t ) désigne le nombre de partitions de i en exactement t parts. On donne aussi une...

Permutations which make transitive groups primitive

Pedro Lopes (2009)

Open Mathematics

In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups...

Polynomial analogues of Ramanujan congruences for Han's hooklength formula

William J. Keith (2013)

Acta Arithmetica

This article considers the eta power ∏ ( 1 - q k ) b - 1 . It is proved that the coefficients of q n / n ! in this expression, as polynomials in b, exhibit equidistribution of the coefficients in the nonzero residue classes mod 5 when n = 5j+4. Other symmetries, as well as symmetries for other primes and prime powers, are proved, and some open questions are raised.

Proof of a conjecture of Hirschhorn and Sellers on overpartitions

William Y. C. Chen, Ernest X. W. Xia (2014)

Acta Arithmetica

Let p̅(n) denote the number of overpartitions of n. It was conjectured by Hirschhorn and Sellers that p̅(40n+35) ≡ 0 (mod 40) for n ≥ 0. Employing 2-dissection formulas of theta functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating function for p̅(40n+35) modulo 5. Using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we prove the congruence p̅(40n+35) ≡ 0 (mod 5) for n ≥ 0. Combining this congruence and the congruence p̅(4n+3) ≡ 0 (mod...

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