Abstract:Inspired by the recent work of Baruah and Ojah, we investigate several partition functions and establish some interesting Ramanujan-type congruences.
“…About further research of arithmetic properties of cubic partitions, overcubic partitions and other colored partitions, some interesting results can be found in [18,21,23,24]. The first author [25] of the present paper generalized Hammond-Lewis birank and gave combinatorial interpretations for some colored partitions.…”
Section: Introductionmentioning
confidence: 80%
“…where γ(n) is defined by ∞ n=0 γ(n)q n = (−q;q 2 ) 4 ∞ (q 2 ;q 2 )∞ . (see [23]). It can also prove the identity γ(7n + 2) ≡ 0 (mod 7).…”
Section: Rank and Crank Analogs For Colored Partitions Congruences Momentioning
confidence: 96%
“…where α(n) is defined by ∞ n=0 α(n)q n = 1 (q;q) 2 ∞ (q 2 ;q 2 )∞ . (see [23]). Suppose is primitive 7th root of unity.…”
Section: Rank and Crank Analogs For Colored Partitions Congruences Momentioning
confidence: 96%
“…In this section, we define statistics that divide the relevant partitions into equinumerous classes and provide the combinatorial interpretation according to [17] for colored partitions congruences modulo 7 given in Toh [21], Zhang and Wang [23].…”
Section: Rank and Crank Analogs For Colored Partitions Congruences Momentioning
We establish some rank and crank analogs for partitions into distinct colors and give combinatorial interpretations for colored partitions such as partitions defined by Toh, Zhang and Wang congruences modulo 5, 7.
“…About further research of arithmetic properties of cubic partitions, overcubic partitions and other colored partitions, some interesting results can be found in [18,21,23,24]. The first author [25] of the present paper generalized Hammond-Lewis birank and gave combinatorial interpretations for some colored partitions.…”
Section: Introductionmentioning
confidence: 80%
“…where γ(n) is defined by ∞ n=0 γ(n)q n = (−q;q 2 ) 4 ∞ (q 2 ;q 2 )∞ . (see [23]). It can also prove the identity γ(7n + 2) ≡ 0 (mod 7).…”
Section: Rank and Crank Analogs For Colored Partitions Congruences Momentioning
confidence: 96%
“…where α(n) is defined by ∞ n=0 α(n)q n = 1 (q;q) 2 ∞ (q 2 ;q 2 )∞ . (see [23]). Suppose is primitive 7th root of unity.…”
Section: Rank and Crank Analogs For Colored Partitions Congruences Momentioning
confidence: 96%
“…In this section, we define statistics that divide the relevant partitions into equinumerous classes and provide the combinatorial interpretation according to [17] for colored partitions congruences modulo 7 given in Toh [21], Zhang and Wang [23].…”
Section: Rank and Crank Analogs For Colored Partitions Congruences Momentioning
We establish some rank and crank analogs for partitions into distinct colors and give combinatorial interpretations for colored partitions such as partitions defined by Toh, Zhang and Wang congruences modulo 5, 7.
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