G.E Andrews et al Res Number Theory (2021) 7:56 https://doi.org/10.1007/s40993-021-00285-7 RESEARCH A Stanley–Elder theorem on Cranks and Frobenius symbols George E Andrews1* , Manosij G Dastidar1 and Thomas Morrill2,3 * Correspondence: gea1@psu.edu.in Pennsylvania State University, University Park, PA 16802, USA Full list of author information is available at the end of the article Partially supported by Simons Foundation Grant 633284 Abstract The Stanley–Elder theorem asserts that the number of j’s in the partitions of n is equal to the number of parts that appear at least j times in a given partition of n, summed over all partitions of n In this paper, we prove that the number of partitions of n with crank > j equals to half the total number of j’s in the Frobenius symbols for n Keywords: Partitions, Cranks, Frobenius symbols, Durfee squares Introduction One of the most charming results in the elementary theory of partitions is the Stanley– Elder theorem Theorem (Stanley–Elder) For each j ≥ the number of j ′ s used in the partitions of n equals the number of parts that occur at least j times in a given partition of n, summed over all the partitions of n Example For n = , j = there are partitions: , + , + 2, + + , + + , + + + , + + + + with 2’s occurring times and the partitions + + , + + 1, + + + 1, + + + + each have one instance of a part occurring twice or more, in total times In [7], R Gilbert provides a complete history of this theorem including the fact that it was originally proved by N.J Fine ([6]; Sec 22) Our object in this paper is to prove a very similar theorem relating the Cranks of partitions to the Frobenius symbols for partitions The surprise of such a result lies in the fact that Cranks seemingly have been historically unrelated to Frobenius symbols; indeed it would be an ominous task to define the crank based on the related Frobenius symbol Since both Cranks and Frobenius symbols are somewhat esoteric, we provide the list of their definitions Definition For a partition π, let l(π) denote the largest part of π, w(π) denote the number of 1’s in π and µ(π) denote the number of parts of π that are larger than w(π) 123 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021 0123456789().,–: volV Content courtesy of Springer Nature, terms of use apply Rights reserved 56 Page of G.E Andrews et al Res Number Theory (2021) 7:56 The crank c(π), is given by: ⎧ ⎨l(π) c(π) = ⎩µ(π) − w(π) if w(π) = 0, if w(π) > Definition The Frobenius symbol is extracted from the Ferrers graph of a partition π as follows: Delete the diagonal of the Ferrers graph If the diagonal was of length j, form the top row of the Frobenius symbol using the rows to the right of the diagonal and respectively form the bottom row from the columns below the diagonal [1, sec 1–3]) Thus if the partition is + + + + then the Ferrers graph is: 421 420 Note that the crank of this partition is −1 = 3, a number difficult to determine directly from the symbol in a general setting Our main theorem is: and correspondingly the Frobenius symbol is Theorem For each j ≥ , the number of partitions of n with Cranks > j equals one half of the number of j ′ s in the Frobenius symbols for the partitions of n Our theorem is an obvious companion to the works [8] of Hopkins, Sellers and Stanton In that paper Theorem states: The number of partitions of n with crank > j equals the number of partitions of n − j with no j in its top row In Sect 2, we shall prove Theorem In Sect 3, we prove a further theorem which is a natural by-product of the proof of Theorem and which has interesting corollaries We conclude with a discussion of further research Proof of main results Recalling the work [8] of Hopkins, Sellers and Stanton mentioned above, we should note that the first part of our proof of Theorem reproves a result from their paper Since our proof differs from theirs, it seems appropriate to include it both for completeness and contrast In [4], M(m, n) is defined to be the number of partitions of n with crank m, and the generating function is given by ∞ ∞ M(m, n)q n z m = n=0 m=−∞ (q; q)∞ (zq; q)∞ (q/z; q)∞ where (A; q)N = (1 − A)(1 − Aq) (1 − Aq N −1 ), and (A; q)∞ = lim (A; q)N N →∞ Content courtesy of Springer Nature, terms of use apply Rights reserved G.E Andrews et al Res Number Theory (2021) 7:56 Page of Hence by [2, p 19, Eq (2.2.5)] ∞ ∞ M(m, n)q n z m = (q; q)∞ n=0 m=−∞ n,m z n−m q n+m (q; q)n (q; q)m (2.1) We now denote by Mj (m, n) the number of partitions of n with crank m > j Note that Mj (m, n) = for m ≤ j Thus we see in (2.1) that in order to have Cranks > j, we must have n − m > j Therefore ∞ ∞ ∞ Mj (m, n)q n z m = (q; q)∞ n=0 m=−∞ m=0 z −m q m (q; q)m ∞ n=m+j+1 = q j+1 z j+1 (q; q)∞ n zn qn (q; q)n zn qn · (q; q)n+j+1 m q 2m (q; q)m (q n+j+2 ; q)m (2.2) Now in Heine’s transform [2], p 19, Cor 2.3 take a = b = 0, t = q , c = q n+j+2 ⎛ ⎞ n qn z ⎠· = q j+1 z j+1 (q; q)∞ ⎝ (q; q)n+j+1 (q n+j+2 ; q)∞ (q ; q)∞ n ⎛ ⎞ m q (m )+m(n+j+2) (q ; q)m (−1) ⎠ ×⎝ (q; q)m m = q j+1 z j+1 (q; q)∞ q j+1 z j+1 = (q; q)∞ = q j+1 z j+1 (q; q)∞ (−1)m q ( zn qn · · m+1 (2.3) )+m(n+j+1) (1 − q m+1 ) m n (−1)m q ( m+1 )+m(j+1) (1 − q m+1 ) · z n q n(m+1) n m (−1)m q ( m+1 )+m(j+1) (1 − q m+1 ) − zq m+1 m (2.4) Setting z = in this last expression we find that the total number of Cranks > j in the partitions of n is generated by: ∞ ∞ Mj (m, n)q n = n=0 m=−∞ = q j+1 (q; q)∞ (q; q)∞ (−1)m q ( m+1 )+m(j+1) m (2.5) (−1)m−1 q ( m+1 )+mj m Now we turn to the generating function for the Frobenius symbols for the partitions of n (cf [1, secs 1–3]) As is shown there, the generating function is [z ](−zq; q)∞ (−1/z; q)∞ (2.6) N where [z j ] ∞ N =0 AN z = Aj The factor (1 + zq j+1 ) produces the possible j in the top row of the Frobenius symbol, and the factor (1 + z −1 q j ) produces the j in the bottom row hence to keep track of the j ′ s we must replace (1 + zq j+1 ) by (1 + yzq j+1 ) Content courtesy of Springer Nature, terms of use apply Rights reserved 56 56 Page of G.E Andrews et al Res Number Theory (2021) 7:56 and replace (1 + z −1 q j ) by (1 + yz −1 q j ) in the infinite product occurring in (2.6), Thus we are to consider [z ] (1 + yzq j+1 )(1 + yz −1 q j ) (−zq; q)∞ (−1/z; q)∞ (1 + zq j+1 )(1 + z −1 q j ) To obtain the generating function that counts the total number of j ′ s we must differentiate with respect to y and set y = Therefore the generating function for the total number of all the j ′ s in the Frobenius symbols for n is given by [z ] (zq j+1 + z −1 q j + 2q 2j+1 ) (−zq; q)∞ (−1/z; q)∞ (1 + zq j+1 )(1 + z −1 q j ) = [z ](1 − (1 − q 2j+1 ) )(−zq; q)∞ (−1/z; q)∞ (1 + zq j+1 )(1 + z −1 q j ) = − [z ](1 − q 2j+1 ) (q; q)∞ ∞ n (n+1 ) n=∞ z q ∞ ∞ (−zq j+1 m ) m=0 (−z −1 j h q) (q; q)∞ h=0 , (2.7) by[2,p.21;Thm.2.8] where (1 ≤ |z| ≤ ) |q| Now the terms with z arise precisely when n = h − m Hence the above is equal to = (1 − q 2j+1 ) − (q; q)∞ (q; q)∞ q (j+1)m+hj+( (1 − q 2j+1 ) ⎝ = − (q; q)∞ (q; q)∞ + m>h − (1 − q 2j+1 ) (q; q)∞ (1 − q 2j+1 ) (q; q)∞ ) (−1)m+h m,h ⎛ = − (q; q)∞ m−h h m ⎞ ⎠ q (j+1)m+hj+( q (j+1)(m+h+1)+hj+( m−h ) (−1)m+h (2.8) m+1 ) (−1)m+1 m+1 ) (−1)m+1 m,h q (j+1)m+(h+m)j+( h+1 ) (−1)h m,h (Next we interchange h and m in the second sum) = (1 − q 2j+1 ) − (q; q)∞ (q; q)∞ q (j+1)(m+h+1)+hj+( m,h (1 − q 2j+1 ) m+1 q (j+1)h+(h+m)j+( ) (−1)m (q; q)∞ m,h ⎡ m+1 ⎣1 − = q ( )+(m+1)(j+1) (−1)m+1 − (q; q)∞ − m (2.9) q( m m+1 ⎤ )+mj (−1)m ⎦ Content courtesy of Springer Nature, terms of use apply Rights reserved G.E Andrews et al Res Number Theory (2021) 7:56 Page of (where we have summed the geometric series with index h) = ⎡ ⎣1 − (q; q)∞ = (q; q)∞ q( m+1 m m (−1) q( )+mj (−1)m − m−1 q( m+1 )+mj m+1 ⎤ )+mj (−1)m ⎦ (2.10) m Comparing the generating functions given in (2.5) and (2.10) we see that Theorem is proved Related results We begin with an assertion that is a restatement of Theorem However with this new formulation we are able to obtain to appealing corollaries Theorem Let π be a partition of n with c(π) = k > Then there is a one-to-one correspondence between π and a set consisting of two occurrences of each of the integers i with ≤ i ≤ k − among all of the parts of the Frobenius symbols for the partitions of n In the following, we denote by M2 (n) the second moments for the Cranksc(π)2 M2 (n) = π ⊢n Corollary M2 (n) = np(n) Remark This result has been proved many times previously Here we have a fairly combinatorial proof Proof We see that M2 (n) = c(π)2 c(π )>0 c(π )−1 (2i + 1) = c(π )>0 i=0 and by Theorem we see that this latter sum adds up each part among all the Frobenius symbols with the “+1” accounting for the contribution from the diagonal in the Ferrers graph), i.e the latter sum is just np(n) ⊓ ⊔ Corollary The sum of the side lengths of all the Durfee squares (equivalently the sums of the lengths of the Frobenius symbols) in the partitions of n equals the sum of all the positive Cranks in the partitions of n Content courtesy of Springer Nature, terms of use apply Rights reserved 56 56 Page of G.E Andrews et al Res Number Theory (2021) 7:56 Proof By Theorem 3, the sum of c(π) over all partitions of n is equal to half the number of parts in the Frobenius symbols corresponding to π Summing over all partitions produces half the number of parts in all Frobenius sybmols for partitions of n, and this is exactly the sum of the Durfee square side lengths of those partitions ⊓ ⊔ Conclusion Prior to the discoveries of this paper and those of [8], there was no reason to suspect that there would be any connection between Cranks and Frobenius symbols So one would hope that there is something combinatorial underlying Theorem that would shed light on this mystery In addition to the relation of this paper to work in [8], it should be noted that the case j = of Theorem has appeared in different guises in the literature (cf [3,5,9]) Author details Pennsylvania State University, University Park, PA 16802, USA, Technische Universität Wien, 1040 Vienna, Austria, Trine University, One University Avenue, Angola, IN 46703, USA Received: 13 May 2021 Accepted: 29 July 2021 Published online: 18 August 2021 References Andrews, G.E.: Generalized Frobenius partitions Mem Am Math Soc 301, 44 (1984) Andrews, G.E.: The Theory of Partitions Addison-Wesley, Reading, 1976 (Revised: Cambridge University Press, Cambridge) (1998) Andrews, G.E.: Concave compositions Electron J Combin P6 (2011) Andrews, G.E., Garvan, F.G., et al.: Dyson’s crank of a partition Bull (New Ser.) Am Math Soc 18(2), 167–171 (1988) Andrews, G.E., Newman, D.: The minimal excludant in integer partitions J Integer Seq 23(2), 20–2 (2020) Fine, N.J.: Basic hypergeometric series and applications 27 American Mathematical Soc (1988) Gilbert, R.A.: A fine rediscovery Am Math Monthly 122(4), 322–331 (2015) Hopkins, B., Sellers, J.A., Stanton, D.: Dyson’s Crank and the Mex of integer partitions In: arXiv preprint arXiv:2009.10873 (2020) Uncu, A.K.: Weighted Rogers–Ramanujan partitions and Dyson crank Ramanujan J 46(2), 579–591 (2018) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations Content courtesy of Springer Nature, terms of use apply Rights reserved Terms and Conditions Springer Nature journal content, brought to you courtesy of Springer Nature Customer Service Center GmbH (“Springer Nature”) Springer Nature supports a reasonable amount of sharing of research papers by authors, subscribers and authorised users (“Users”), for smallscale personal, non-commercial use provided that all copyright, trade and service marks and other proprietary notices are maintained By accessing, sharing, receiving or otherwise using the Springer Nature journal content you agree to these terms of use (“Terms”) For these purposes, Springer Nature considers academic use (by researchers and students) to be non-commercial These Terms are supplementary and will apply in addition to any applicable website terms and conditions, a relevant site licence or a personal subscription These Terms will prevail over any conflict or ambiguity with regards to the relevant terms, a site licence or a personal subscription (to the extent of the conflict or ambiguity only) For Creative Commons-licensed articles, the terms of the Creative Commons license used will apply We collect and use personal data to provide access to the Springer Nature journal content We may also use these personal data internally within ResearchGate and Springer Nature and as agreed share it, in an anonymised way, for purposes of tracking, analysis and reporting We will not otherwise disclose your personal data outside the ResearchGate or the Springer Nature group of companies unless we have your permission as detailed in the Privacy Policy While Users may use the Springer Nature journal content for small scale, personal non-commercial use, it is important to note that Users may not: use such content for the purpose of providing other users with access on a regular or large scale basis or as a means to circumvent access control; 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