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a _ partition _ of a positive integer @xmath3 is a non - increasing sequence of positive integers whose sum is @xmath3 ; the number of distinct partitions of @xmath3 is traditionally denoted by @xmath4 . a _ sequence _ ( or _ run _ ) in a partition is any subsequence of consecutive integers that appear as parts . integer partitions without sequences were first studied by macmahon in @xcite . he described their relationship to partitions with repeated parts , and also determined their generating series . these partitions were studied more recently by holroyd , liggett , and romik in @xcite , where the authors introduced the general family of _ partitions without @xmath0-sequences _ for any @xmath5 , in which no @xmath0 consecutive integers may all appear as parts . the number of partitions of @xmath3 without @xmath0-sequences of @xmath3 is denoted by @xmath6 , and the generating function is defined as @xmath7 these partitions were also studied by andrews @xcite , who found a ( double ) @xmath1-hypergeometric series expansion for the generating function . before giving this series , we record the definition for the @xmath1-pochhammer symbol , which is given by @xmath8    andrews showed that @xmath9    both of these earlier papers also addressed the asymptotic behavior of @xmath6 as @xmath10 . holroyd , liggett , and romik @xcite showed the asymptotic expression @xmath11 andrews @xcite substantially improved this result in the case @xmath12 , proving the asymptotic expression @xmath13 his proof relies on a beautiful and surprising relationship between @xmath14 and one of ramanujan s famous mock theta functions ; we will further discuss this connection below . the above asymptotic expression is not stated as such in andrews paper . he instead focused on the asymptotics of the generating series @xmath14 as @xmath15 . however , his results directly imply upon applying a tauberian theorem ( cf . section [ s : overpartnasymp ] of this paper ) . recently was greatly refined by the first and third authors of the present paper in @xcite . specifically , the latter paper introduced a generalization of the circle method in order to prove a series expansion for @xmath16 involving kloosterman sums , bessel functions , and principal value integrals of modified bessel functions . in this paper we study a related family of overpartitions . as introduced by corteel and lovejoy in @xcite , an overpartition is a partition in which the last occurrence of each part may be overlined . the number of overpartitions of size @xmath3 is denoted by @xmath17 . an overpartition is said to have a gap at @xmath18 if there are no parts of size @xmath18 . we define lower @xmath0-run overpartitions to be those overpartitions in which any overlined part must occur within a run of exactly @xmath0 consecutive overlined parts that terminates below with a gap . more precisely , this means that if some part @xmath19 is overlined , then there is an integer @xmath20 with @xmath21 $ ] such that each of the @xmath0 overlined parts @xmath22 appear ( perhaps together with non - overlined versions ) , while no part @xmath20 ( overlined or otherwise ) appears , and no overlined part @xmath23 appears . there is a simple bijection between lower @xmath0-run overpartitions and upper @xmath0-run overpartitions , which are defined analogously but with the conditions on parts @xmath20 and @xmath24 reversed ( see section [ s : combinatorial ] ) . denote the number of lower @xmath0-run overpartitions of size @xmath3 by @xmath25 . the lower @xmath26-run overpartitions of size @xmath27 are @xmath28 together with the @xmath29 partitions of @xmath27 , so @xmath30 . the generating function for lower @xmath0-run overpartitions is denoted by @xmath31 our first result is analogous to andrews double - series generating function for partitions without @xmath0-sequences . [ t : gbarq ] for @xmath32 , @xmath33    our next result is an asymptotic expression for lower @xmath0-run overpartitions that is much stronger than the logarithmic expression in . [ t : pk ] as @xmath34 , @xmath35    interestingly , our techniques do not apply to the case of partitions without sequences , despite the similarity of and theorem [ t : gbarq ] . we next focus particularly on certain special cases that are related to ramanujan s mock theta functions . andrews @xcite showed that the generating function for partitions without @xmath26-sequences may be written as @xmath36 where @xmath37 which is one of ramanujan s third - order mock theta functions . ramanujan originally introduced the mock theta functions by listing a small collection of examples in his last letter to hardy @xcite . he justified his own interest by observing their striking asymptotic properties and near - symmetries under modular transformations . andrews @xcite used some of these properties in order to determine the asymptotic behavior of @xmath14 as @xmath38 , which then implies . the general theory of mock theta functions has also recently seen great advancements , as zwegers doctoral thesis @xcite has led to a proper understanding of the relationship to automorphic forms @xcite .    the case @xmath39 of lower @xmath0-run overpartitions is similarly related to another of ramanujan s third - order mock theta functions from @xcite . in particular , the mock theta function @xmath40 appears in the following expression for the generating function of lower @xmath2-run overpartitions . [ p : gbar1 ] for @xmath32 , @xmath41    note that the @xmath39 case of theorem [ t : pk ] states that @xmath42    the remainder of the paper is structured as follows . in section [ s : overpartitions ] , we consider basic combinatorial properties of @xmath0-run overpartitions and derive their generating series . in section [ s : constant ] , we apply the constant term method to determine the asymptotic behavior of the generating series . finally , in section [ s : overpartnasymp ] , we prove the asymptotic expression for @xmath0-run overpartitions found in theorem [ t : pk ] . in this section we denote the number of lower @xmath0-run overpartitions of @xmath3 by @xmath43 . we also recall the definition upper @xmath0-run overpartitions as overpartitions in which the overlined parts must occur in consecutive runs of the form @xmath44 , with no overlined parts @xmath45 , and no parts of any kind of size @xmath24 . the number of upper @xmath0-run overpartitions of size @xmath3 is denoted by @xmath46 . our first observation on @xmath0-run overpartitions is that the lower and upper definitions are in bijective correspondence . [ p : op-+ ] for all @xmath47 and @xmath48 , @xmath49    we construct a simple bijection between the two sets of overpartitions . in a lower @xmath0-run overpartition , any run of @xmath0 overlined parts occurs at the lower end of some block of consecutive parts , surrounded on both sides by gaps . that is , for some @xmath20 and some @xmath50 , all the parts @xmath51 appear ( perhaps together with further non - overlined copies of these same values ) , with gaps at @xmath20 and @xmath52 . we can form a new overpartition by shifting the @xmath0 overlines to the upper end of the block , i.e. ,  replacing the above parts with @xmath53 applying this transformation to every run of @xmath0 overlined parts results in an upper @xmath0-run overpartition , and this map is clearly bijective . we henceforth write simply @xmath54 for @xmath55 , and abbreviate the term `` lower @xmath0-run overpartition '' to @xmath0-run overpartition . it will be convenient to work with both the upper and lower definitions in different contexts . in view of proposition  [ p : op-+ ] they are interchangeable for purposes of enumeration . we have the following monotonicity properties for @xmath0-run overpartitions . [ p : mono ] for any @xmath56 and @xmath48 ,    1 . [ p : mono : n ] @xmath57 2 . [ p : mono : k ] @xmath58 . [ p : mono : n ] given an upper @xmath0-run overpartition of @xmath3 , adding a non - overlined part @xmath2 clearly gives an upper @xmath0-run overpartition of @xmath59 , and this map is injective . [ p : mono : k ] we construct an injection from lower @xmath60-run overpartitions to lower @xmath0-run overpartitions . if @xmath61 is a @xmath60-run overpartition and @xmath62 is a @xmath60-run , then there are no parts of size @xmath20 , and there can only be non - overlined parts of size @xmath63 . removing the overline on @xmath23 results in a @xmath0-run satisfying the lower @xmath0-run condition , and applying this to every run gives the required injection . we note that the partitions without sequences studied in @xcite satisfy the opposite inequality to that in proposition [ p : mono ] [ p : mono : k ] : @xmath64 2 .   for any fixed @xmath3 , we have @xmath65 for sufficiently large @xmath0 .      in this section we consider the two - variable generating functions @xmath66 where @xmath67 is defined to be the number of @xmath0-run overpartitions of size @xmath3 and exactly @xmath68 parts . the main result in this section is a double series expansion for this generating function , which is proven from a @xmath1-difference equation for @xmath69 . [ t : gbarxq ] for @xmath32 , @xmath70    the generating function @xmath71 satisfies the @xmath1-difference equation @xmath72 which follows from the definition of ( upper ) @xmath0-run overpartitions by separating according to their smallest parts . the first term corresponds to the overpartitions without an overlined @xmath73 , in which case there is no restriction on subsequent parts ; the second term corresponds to overpartitions that do have the part @xmath73 , and thus also have each of the parts @xmath74 ( and possibly non - overlined parts of these sizes as well ) , followed by no parts of size @xmath75 . this recurrence is easier to solve after `` re - normalizing '' the equation , which is achieved by setting @xmath76 equation then implies that this normalized function satisfies the @xmath1-difference equation @xmath77 we next write this as a series in @xmath78 , where the coefficients are polynomials in @xmath1 , @xmath79 equation is equivalent to the statement that for all @xmath18 , @xmath80    by , the theorem statement is equivalent to verifying that @xmath81 is equal to the double series @xmath82 it is sufficient to show that this double series satisfies the @xmath1-difference equation , while also checking that the @xmath75 initial @xmath78-powers are compatible . denote the @xmath83 coefficient in the double series by @xmath84 ; it is clear by definition that @xmath85 and that @xmath86 , and finally @xmath87 note that @xmath0-run overpartitions with at most @xmath88 parts can contain only non - overlined parts , and thus @xmath89 the first term gives all overpartitions with no overlined parts , and the second term gives all overpartitions that consist of a single @xmath0-run of overlined parts . thus @xmath90 and the initial terms @xmath91 @xmath92 agree with those given for @xmath93 above . now we verify that the @xmath93 satisfy . temporarily writing @xmath94 we then have @xmath95 and thus @xmath96 \\ & = \sum_{\substack{r , s \geq 0 : \\ kr + ( k+1)s = m - k - 1 } } a_{r , s } \cdot q^{{\textstyle\frac{k(k+1)(2r + 2s + 1)}{2}}+ ( k+1)(s+1 ) } \\ & \qquad+ \sum_{\substack{r , s \geq 0 : \\ kr + ( k+1)s = m - k } } a_{r , s } \cdot q^{{\textstyle \frac{k(k+1)(2r + 2s + 1)}{2 } } + ( k+1)s } \\ & = q^{m(k+1 ) - \frac{k(k+1)}{2 } }   \left(\widetilde{\lambda}_{m - k } - \widetilde{\lambda}_{m - k-1}\right).\end{aligned}\ ] ] the two sums in the second line were shifted by @xmath97 and @xmath98 , respectively . this completes the proof of the double series formula .      in this section we prove theorem  [ p : gbar1 ] . recall the earlier definition of ramanujan s third - order mock theta function @xmath99 this was proven by fine ( see equation ( 26.32 ) of @xcite ) to have the equivalent form @xmath100    we first describe a family of overpartitions that are in bijective correspondence with lower @xmath2-run overpartitions . the bijection will be the _ conjugation _ involution . an overpartition may be represented by a _ ferrer s diagram _ that lists each part as a left - justified row of dots , with the parts listed in decreasing order from top to bottom . furthermore , since an overlined part is the last occurrence of any part size , we distinguish it by marking the corresponding bottom - right `` corner '' . the conjugation map then simply interchanges the rows and columns of such a diagram . under overpartition conjugation , lower @xmath2-run overpartitions map to overpartitions in which overlined parts may not occur alone , except for possibly in the largest part . in other words , such overpartitions have the property that if @xmath101 and @xmath18 is not the largest part size , then @xmath102 as well . see figure [ f : conj ] for an example of the conjugate of a @xmath2-run overpartition .    by distinguishing the largest part , the generating function for such partitions is clearly @xmath103 which is easily seen to be equal to the second factor in . throughout this section we view @xmath1-series as fourier expansions . in fact , we will mainly focus on real values of @xmath104 , so we write @xmath105 with @xmath106 consider the double hypergeometric @xmath1-series from theorem [ t : gbarq ] , which we denote as @xmath107    [ p : hkq ] as @xmath108 , @xmath109    the proof follows the _ constant - term _ method , which begins with the observation that @xmath110 can be written as a ( laurent ) coefficient of a two - variable series , namely @xmath111}\left(\ \sum_{n\in{{\mathbb{z}}}}q^{\frac{k(k+1)n^2}{2}}x^{-n } \sum_{r\geq 0}\frac{x^r}{\big(q^k ; q^k\big)_r } \sum_{s\geq 0}\frac{(-1)^s x^s q^{\frac{(k+1)s(s+1)}{2}}}{\big(q^{k+1 } ; q^{k+1}\big)_s } \right).\ ] ] as indicated , the series on the right side of factorizes into three sums ( in @xmath3 , @xmath112 , and @xmath113 ) , and we now express each of the three factors in terms of known number - theoretic functions . note that all of the summations are convergent for @xmath114 . the first factor is @xmath115 , where @xmath116 is the _ jacobi theta function_.    the second and third summations can both be expressed as infinite products via the following instances of the @xmath1-binomial theorem @xcite : @xmath117 we further rewrite these terms using the _ quantum dilogarithm _ ; this function is defined by @xmath118    the expression in can now be written as @xmath119\bigg ( \theta\left(q^{\frac{k(k+1)}2};x\right)\ , \exp\bigg(\operatorname{li}_2\left(x;q^k\right)-\operatorname{li}_2\left(x q^{k+1 } ; q^{k+1}\right)\bigg)\bigg).\ ] ] applying cauchy s theorem , we recover the @xmath120-coefficient as @xmath121+ic } \theta\left(q^{\frac{k(k+1)}2};e^{2 \pi i u}\right)\ ,   \exp\bigg(\operatorname{li}_2\left(e^{2 \pi i u};q^k\right)-   \operatorname{li}_2\left(e^{2 \pi i u } q^{k+1 } ; q^{k+1}\right)\bigg ) du,\end{aligned}\ ] ] where @xmath122 is a real constant that will be specified later . the theta function may be transformed using the standard poisson summation formula @xcite , resulting in @xmath123 our integral then transforms to @xmath124+ic } \sum_{n\in{{\mathbb{z } } } } \exp\left(-\frac{2\pi^2 ( n+u)^2}{\varepsilon k(k+1 ) } + \operatorname{li}_2\left(e^{2 \pi i u};q^k\right)-\operatorname{li}_2\left(e^{2 \pi i u } q^{k+1 } ; q^{k+1}\right)\right ) du \notag \\ \label{e : hkqint } & = \sqrt{\frac{2\pi^2}{\varepsilon k(k+1 ) } } \int_{{{\mathbb{r}}}+ic } \exp\left ( -\frac{2\pi u^2}{\varepsilon k(k+1 ) } + \operatorname{li}_2\left(e^{2\pi iu } ; e^{-k\varepsilon}\right ) -\operatorname{li}_2\left(e^{2\pi iu-(k+1)\varepsilon } ; e^{-(k+1)\varepsilon}\right)\right ) du.\end{aligned}\ ] ]    we now apply the _ stationary phase _ method to this integral , which is generally useful for determining the asymptotic behavior of integrals of the form @xmath125 as @xmath108 . if @xmath126 has a critical point at @xmath127 , and both @xmath126 and @xmath128 have taylor expansions around this point , then the dominant asymptotic term of the integral can be described in terms of @xmath129 and @xmath130 . in order to apply the stationary phase method to , we must first rewrite the integrand in this shape in order to identify the dominant asymptotic terms . if @xmath131 and @xmath132 then the laurent expansion of the quantum dilogarithm begins @xmath133 uniformly in @xmath78 as @xmath134 . here @xmath135 is the standard dilogarithm function . using , we study the argument of the exponential in the integrand of , namely @xmath136 in particular , we consider the laurent expansion of and denote the coefficient of the ( leading ) @xmath137 term by @xmath138    we proceed by determining the critical point(s ) of @xmath126 . its derivative is @xmath139 and the critical point therefore occurs at @xmath140 we denote this critical value of @xmath141 by @xmath142 , which also determines the most appropriate height of the contour ( specifically , @xmath143 ) . at the critical value , the function @xmath126 evaluates to @xmath144 here we used the functional equation @xcite @xmath145 by definition @xmath146 , but the second derivative still makes a contribution to the overall asymptotic behavior , so we calculate @xmath147 we can therefore compute the first terms in the taylor expansion around @xmath148 of . in particular , using the change of variable @xmath149 , this taylor expansion is @xmath150    plugging in the expansions and to the integral , the contour may then be shifted to the real axis , leading finally to the evaluation @xmath151 in this section we use the asymptotic behavior of generating series in order to determine the asymptotic behavior of @xmath25 as @xmath10 . as before , we write @xmath152 . recall that @xmath153 ingham s tauberian theorem relates the asymptotic behavior of such a series to its coefficients . the following result is a special case of theorem 1 in @xcite . [ t : ingham ] let @xmath154 be a power series with real nonnegative coefficients and radius of convergence equal to @xmath2 . if there exist @xmath155 , @xmath156 such that @xmath157 as @xmath158 , then @xmath159 as @xmath160 . the modular inversion formula for dedekind s eta - function ( page 121 , proposition 14 of @xcite ) states that @xmath161 this implies that @xmath162 @xmath163 combined with proposition [ p : hkq ] , this implies that as @xmath164 @xmath165 note that the coefficients of this @xmath1-series are @xmath166 . applying theorem [ t : ingham ] with @xmath167 then gives the stated asymptotic formula for @xmath25 by telescoping .
in this paper we introduce @xmath0-run overpartitions as natural analogs to partitions without @xmath0-sequences , which were first defined and studied by holroyd , liggett , and romik . following their work as well as that of andrews , 
 we prove a number of results for @xmath0-run overpartitions , beginning with a double summation @xmath1-hypergeometric series representation for the generating functions . in the special case of @xmath2-run overpartitions 
 we further relate the generating function to one of ramanujan s mock theta functions . 
 finally , we describe the relationship between @xmath0-run overpartitions and certain sequences of random events , and use probabilistic estimates in order to determine the asymptotic growth behavior of the number of @xmath0-run overpartitions of size @xmath3 .