A stanley elder theorem on cranks and fr

Dastidar1and 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

R E S E A R C H

A Stanley–Elder theorem on Cranks and

Frobenius symbols

George E Andrews1*, Manosij G Dastidar1and 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

1 Introduction

One of the most charming results in the elementary theory of partitions is the Stanley– Elder theorem

Theorem 1 (Stanley–Elder) For each j ≥ 1 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 = 5 , j = 2 there are 7 partitions: 5 , 4 + 1 , 3 + 2, 3 + 1 + 1 , 2 + 2 + 1 ,

2 + 1 + 1 + 1 , 1 + 1 + 1 + 1 + 1 with 2’s occurring 4 times and the partitions 3 + 1 + 1 , 2 + 2 + 1, 2 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1 each have one instance of a part occurring twice or more, in total 4 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 1 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.

The crank c(π ), is given by:

c(π ) =

⎧

⎨

⎩

l(π ) if w(π ) = 0, µ(π ) − w(π ) if w(π ) > 0

Definition 2 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 5 + 4 + 4 + 2 + 1 then the Ferrers graph is:

and correspondingly the Frobenius symbol is



4 2 1

4 2 0

 Note that the crank of this partition is 4 −1 = 3, a number difficult to determine directly from the symbol in a general setting

Our main theorem is:

Theorem 2 For each j ≥0 , 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 8 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 Theorem2 In Sect.3, we prove a further theorem which is a

natural by-product of the proof of Theorem2and which has interesting corollaries We

conclude with a discussion of further research

2 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 Theorem2reproves 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

∞



n=0

∞



m=−∞

M(m, n)qnzm= (q; q)∞

(zq; q)∞(q/z; q)∞

where (A; q)N =(1 − A)(1 − Aq) (1 − AqN −1), and (A; q)∞= lim

N →∞(A; q)N

Hence by [2, p 19, Eq (2.2.5)]

∞



n=0

∞



m=−∞

M(m, n)qnzm=(q; q)∞



n,m0

zn−mqn+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) = 0 for m ≤ j Thus we see in (2.1) that in order to have Cranks > j, we must

have n − m > j Therefore

∞



n=0

∞



m=−∞

Mj(m, n)qnzm=(q; q)∞

∞



m=0

z−mqm (q; q)m

∞



n=m+j+1

znqn (q; q)n

=qj+1zj+1(q; q)∞



n0

znqn (q; q)n+j+1 ·



m0

q2m (q; q)m(qn+j+2; q)m

(2.2) Now in Heine’s transform [2], p 19, Cor 2.3 take a = b = 0, t = q2, c = qn+j+2

=qj+1zj+1(q; q)∞

⎛

⎝



n0

znqn (q; q)n+j+1

1 (qn+j+2; q)∞(q2; q)∞

⎞

⎠ ·

×

⎛

⎝



m0

(−1)mq(m2)+ m(n+j+2)(q2; q)m

(q; q)m

⎞

⎠

= qj+1zj+1 (q; q)∞

· n0

znqn· m0 (−1)mq(m+1)+m(n+j+1)(1 − qm+1)

= qj+1zj+1 (q; q)∞



m0

(−1)mq(m+1)+m(j+1)(1 − qm+1) ·

n0

znqn(m+1)

(2.3)

=qj+1zj+1 (q; q)∞



m0

(−1)mq(m+1)+m(j+1)(1 − qm+1)

Setting z = 1 in this last expression we find that the total number of Cranks > j in the

partitions of n is generated by:

∞



n=0

∞



m=−∞

Mj(m, n)qn= qj+1

(q; q)∞



m0 (−1)mq(m+1)+m(j+1)

(q; q)∞



m1 (−1)m−1q(m+1)+mj

(2.5)

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

where [zj]∞

N =0ANzN =Aj The factor (1 + zqj+1) produces the possible j in the top row of the Frobenius symbol, and

the factor (1 + z−1qj) produces the j in the bottom row

hence to keep track of the j′swe must replace (1 + zqj+1) by (1 + yzqj+1)

and replace (1 + z− 1qj) by (1 + yz− 1qj) in the infinite product occurring in (2.6),

Thus we are to consider

[z0](1 + yzq

j+1)(1 + yz−1qj) (1 + zqj+1)(1 + z− 1qj) (−zq; q)∞(−1/z; q)∞.

To obtain the generating function that counts the total number of j′swe must differen-tiate with respect to y and set y = 1

Therefore the generating function for the total number of all the j′

sin the Frobenius symbols for n is given by

[z0](zq j+1+z−1qj+2q2j+1) (1 + zqj+1)(1 + z−1qj) (−zq; q)∞(−1/z; q)∞

=[z0](1 − (1 − q

2j+1) (1 + zqj+1)(1 + z−1qj))(−zq; q)∞(−1/z; q)∞

(q; q)∞

−[z0](1 − q2j+1)

∞



m=0 (−zqj+1)m

∞



h=0 (−z−1qj)h

∞ n=∞znq(n+1) (q; q)∞

, by[2,p.21;Thm.2.8]

where (1 ≤ |z| ≤ 1

|q|).

(2.7)

Now the terms with z0arise precisely when n = h − m Hence the above is equal to

(q; q)∞

−(1 − q2j+1) (q; q)∞



m,h0

q(j+1)m+hj+(m−h2 )(−1)m+h

(q; q)∞

−(1 − q2j+1) (q; q)∞

⎛

⎝



m>h0

hm0

⎞

⎠q(j+1)m+hj+(m−h2 )(−1)m+h

(q; q)∞

−(1 − q2j+1) (q; q)∞



m,h0

q(j+1)(m+h+1)+hj+(m+1)(−1)m+1

− (1 − q2j+1) (q; q)∞



m,h0

q(j+1)m+(h+m)j+(h+1)(−1)h

(2.8)

(Next we interchange h and m in the second sum)

(q; q)∞

−(1 − q2j+1) (q; q)∞



m,h0

q(j+1)(m+h+1)+hj+(m+1)(−1)m+1

−(1 − q2j+1) (q; q)∞



m,h0

q(j+1)h+(h+m)j+(m+1)(−1)m

(q; q)∞

⎡

⎣1 − m0

q(m+1)+(m+1)(j+1)(−1)m+1− 

m0 q(m+1)+mj(−1)m

⎤

⎦

(2.9)

(where we have summed the geometric series with index h)

(q; q)∞

⎡

⎣1 − m1

q(m+12 )+ mj(−1)m− 

m0 q(m+12 )+ mj(−1)m

⎤

⎦

(q; q)∞



m1 (−1)m−1q(m+1)+mj

(2.10)

Comparing the generating functions given in (2.5) and (2.10) we see that Theorem2is proved

3 Related results

We begin with an assertion that is a restatement of Theorem2 However with this new

formulation we are able to obtain to appealing corollaries

Theorem 3 Let π be a partition of n with c(π ) = k > 0 Then there is a one-to-one

correspondence between π and a set consisting of two occurrences of each of the integers i

with0 ≤ i ≤ k − 1 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

Cranks-M2(n) =

π ⊢n c(π )2

Corollary 4

1

2M2(n) = np(n) Remark This result has been proved many times previously Here we have a fairly

com-binatorial proof

Proof We see that

1

2M2(n) =



c(π )>0 c(π )2

c(π )>0

c(π )−1



i=0 (2i + 1)

and by Theorem3we 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

Corollary 5

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

Proof By Theorem3, 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 ⊓

4 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 Theorem2that 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 =0 of Theorem1has 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

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5. Andrews, G.E., Newman, D.: The minimal excludant in integer partitions J Integer Seq 23(2), 20–2 (2020)

6 Fine, N.J.: Basic hypergeometric series and applications 27 American Mathematical Soc (1988)

7. Gilbert, R.A.: A fine rediscovery Am Math Monthly 122(4), 322–331 (2015)

8 Hopkins, B., Sellers, J.A., Stanton, D.: Dyson’s Crank and the Mex of integer partitions In: arXiv preprint arXiv:2009.10873

(2020)

9. Uncu, A.K.: Weighted Rogers–Ramanujan partitions and Dyson crank Ramanujan J 46(2), 579–591 (2018)

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These Terms are supplementary and will apply in addition to any applicable website terms and conditions,... partitions In: arXiv preprint arXiv:2009.10873

(2020)

9. Uncu, A. K.: Weighted Rogers–Ramanujan partitions and Dyson crank Ramanujan