SYMMETRY OF k-MARKED DURFEE SYMBOLS

Boulet, Cilanne; Kurşungöz, Kağan

doi:10.1142/s1793042111003971

Cited by 5 publications

(11 citation statements)

References 4 publications

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“…To prove the above two interpretations, we also need the following symmetric property given by Andrews [1]. Boulet and Kursungoz [5] found a combinatorial proof of this fact. (2.2)…”

Section: Combinatorial Interpretationsmentioning

confidence: 99%

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On the Positive Moments of Ranks of Partitions

Chen

Shen

2014

Electron. J. Combin.

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By introducing k-marked Durfee symbols, Andrews found a combinatorial interpretation of 2k-th symmetrized moment η 2k (n) of ranks of partitions of n in terms of (k + 1)-marked Durfee symbols of n. In this paper, we consider the k-th symmetrized positive momentη k (n) of ranks of partitions of n which is defined as the truncated sum over positive ranks of partitions of n. As combintorial interpretations ofη 2k (n) and η 2k−1 (n), we show that for fixed k and i with 1 ≤ i ≤ k + 1,η 2k−1 (n) equals the number of (k + 1)-marked Durfee symbols of n with the i-th rank being zero andη 2k (n) equals the number of (k + 1)-marked Durfee symbols of n with the i-th rank being positive. The interpretations ofη 2k−1 (n) andη 2k (n) also imply the interpretation of η 2k (n) given by Andrews since η 2k (n) equalsη 2k−1 (n) plus twice ofη 2k (n). Moreover, we obtain the generating functions ofη 2k (n) andη 2k−1 (n).

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“…To prove the above two interpretations, we also need the following symmetric property given by Andrews [1]. Boulet and Kursungoz [5] found a combinatorial proof of this fact. (2.2)…”

Section: Combinatorial Interpretationsmentioning

confidence: 99%

“…η 1(5) η 2 (5) η 2 (5)1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 2 1 1 1 2 1 1 2 1 2 1 1 1 1 1 1 2 1 2 1 1 1 1 1 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 2 1 2 1 1 2 1 1 1 1 1 2 1 1 2 1 1 1 2 1 1 1 1 1 2…”

mentioning

confidence: 99%

On the Positive Moments of Ranks of Partitions

Chen

Shen

2014

Electron. J. Combin.

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“…The referee has kindly pointed out that Kagan Kursungoz has also done some studies on the combinatorics of k-marked Durfee symbols in his doctorial thesis [17] and in a paper joint with Boulet [8]. An alternative definition of k-marked Durfee symbols is given in [17].…”

Section: Theorem 13 (Andrews) the Number Of Ordinary Partitions Of mentioning

confidence: 99%

The combinatorics of $k$-marked Durfee symbols

2011

Trans. Amer. Math. Soc.

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Abstract. George E. Andrews recently introduced k-marked Durfee symbols which are connected to moments of Dyson's rank. By these connections, Andrews deduced their generating functions and some combinatorial properties and left their purely combinatorial proofs as open problems. The primary goal of this article is to provide combinatorial proofs in answer to Andrews' request. We obtain a partition identity, which gives a relation between k-marked Durfee symbols and Durfee symbols by constructing bijections, and all identities on k-marked Durfee symbols given by Andrews could follow from this identity. In a similar manner, we also prove the identities due to Andrews on k-marked odd Durfee symbols combinatorially, which resemble ordinary k-marked Durfee symbols with a modified subscript and with odd numbers as entries.

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“…The method employed is to define equivalence classes of Durfee symbols of a given number, and to consider the possible ways to make the symbols in an equivalence class into k-marked Durfee symbols. Part of the results presented in §3 appears in [4], where the authors use the same alternative characterization of k-marked Durfee symbols, but more direct combinatorial methods. In particular, they present a bijection, and a sieve to establish the symmetry, and the relation to ordinary Durfee symbols of the k-marked ones.…”

Section: 1 4 2 2mentioning

confidence: 99%

“…The top row is obtained by reading the conjugate partition of the smaller partition to the right of the Durfee square (recording the columns instead of rows), and the bottom row by reading the smaller partition below the Durfee square. Thus, the partition 36 = 9 + 7 + 7 + 5 + 4 + 2 + 2 has Durfee symbol 4…”

mentioning

confidence: 99%