## On self-conjugate split $$n$$-color partitions.(English)Zbl 1495.05015

Summary: Analogous to the definition of self-conjugate $$n$$-color partitions, we introduce here self-conjugate split $$n$$-color partitions. The self-conjugate split $$n$$-color partitions arise as a modification of another, existing class of partitions. In addition to the generating function and recurrence relation of self-conjugate split $$n$$-color partitions, we find several combinatorial identities which associate these partitions with other combinatorial structures. We give a bijection from the set of split $$n$$-color partitions of a positive integer $$\nu$$ onto that of partitions of $$\nu$$ with “$$\begin{pmatrix} n+1 \\ 2 \end{pmatrix}$$ copies of $$n$$”. Moreover, an explicit bijection between the set of restricted self-conjugate split $$n$$-color partitions of $$\nu$$ and the set of restricted $$n$$-color partitions of $$\nu$$ has been constructed. Some results involving new restricted split $$n$$-color partition functions are also obtained.

### MSC:

 05A15 Exact enumeration problems, generating functions 05A17 Combinatorial aspects of partitions of integers 05A19 Combinatorial identities, bijective combinatorics 11P81 Elementary theory of partitions
Full Text:

### References:

 [1] Agarwal, AK, Rogers-Ramanujan identities for $$n$$-color partitions, J. Number Theory, 28, 299-305 (1988) · Zbl 0641.10012 [2] Agarwal, AK, New combinatorial interpretations of two analytic identities, Proc. Am. Math. Soc., 107, 2, 561-567 (1989) · Zbl 0683.05003 [3] Agarwal, AK, q-functional equations and some partition identities, combinatorics and theoretical computer science (Washington, DC 1989), Discrete Appl. Math., 34, 1-3, 17-26 (1991) · Zbl 0751.05008 [4] Agarwal, AK, New classes of infinite 3-way partition identities, ARS Comb., 44, 33-54 (1996) · Zbl 0888.11040 [5] Agarwal, AK, A note on self-conjugate $$n$$-color partitions, Int. J. Math. Math. Sci., 21, 4, 694 (1998) · Zbl 0912.05001 [6] Agarwal, AK, $$n$$-color compositions, Indian J. Pure Appl. Math., 31, 11, 1421-1427 (2000) · Zbl 0965.05018 [7] Agarwal, AK, Ramanujan congruences for $$n$$-color partitions, Math. Student, 70, 1-4, 199-204 (2001) · Zbl 1132.11363 [8] Agarwal, AK, $$n$$-color analogues of Gaussian polynomials, ARS Comb., 61, 97-117 (2001) · Zbl 1071.05504 [9] Agarwal, AK; Andrews, GE, Rogers-Ramanujan identities for partitions with “N copies of N”, J. Combin. Theory Ser. A, 45, 1, 40-49 (1987) · Zbl 0618.05003 [10] Agarwal, AK; Balasubramanian, R., $$n$$-color partitions with weighted differences equal to minus two, Int. J. Math. Math. Sci., 20, 4, 759-768 (1997) · Zbl 0895.05005 [11] Agarwal, AK; Sachdeva, R., Basic series identities and combinatorics, Ramanujan J., 42, 725-746 (2017) · Zbl 1359.05005 [12] Agarwal, A.K., Sood, G.: Split $$(n+t)$$-color partitions and Gordon-McIntosh eight order mock theta functions. Electron J. Combin. 21(2) (2014), Paper # P2.46 · Zbl 1300.05025 [13] Anand, S.; Agarwal, AK, On some restricted plane partition functions, Utilitas Math., 74, 121-130 (2007) · Zbl 1196.05009 [14] Andrews, GE, The Thoery of Partitions. Encyclopedia of Mathematics and its Applications (1986), Reading, MA: Addison-Wesley, Reading, MA [15] Connor, WG, Partition theorems related to some identities of Rogers and Watson, Trans. Am. Math. Soc., 214, 95-111 (1975) · Zbl 0313.10012 [16] Gordon, B., A combinatorial generalization of the Rogers-Ramanujan identities, Am. J. Math., 83, 393-399 (1961) · Zbl 0100.27303 [17] Gordon, B., Some continued fractions of the Rogers-Ramanujan type, Duke J. Math., 32, 741-748 (1965) · Zbl 0178.33404 [18] Goyal, M., An analogue of Euler’s identity and split perfect partitions, Bull. Aust. Math. Soc., 99, 3, 353-361 (2019) · Zbl 1414.05026 [19] Goyal, M.; Agarwal, AK, On a new class of combinatorial identities, ARS Comb., 127, 65-77 (2016) · Zbl 1413.05024 [20] Hirschhorn, MD, Some partition theorems of the Rogers-Ramanujan type, J. Combin. Theory Ser. A, 27, 1, 33-37 (1979) · Zbl 0418.10015 [21] MacMahon, PA, Combinatory Analysis, Two Volumes (Bound as One) (1960), New York: Chelsea Publishing Co., New York [22] MacMahon, P.A.: In: Andrews, G.E. (ed.) Collected Papers, vol. 1. M.I.T. Press, Cambridge, MA, London, England (1978) · Zbl 0557.01015 [23] Subbarao, MV; Agarwal, AK, Further theorems of Rogers-Ramanujan type, Can. Math. Bull., 31, 2, 210-214 (1988) · Zbl 0615.10020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.