EasyChair Preprint. Computation of Some Integer Sequences in Maple

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1 EasyChair Preprit 4 Coputatio of Soe Iteger Sequeces i Maple W.L. Fa, David J. Jeffrey ad Eri Posta EasyChair preprits are iteded for rapid disseiatio of research results ad are itegrated with the rest of EasyChair. Noveber 20, 2017

2 Coputatio of Soe Iteger Sequeces i Maple W.L. Fa 1, D.J. Jeffrey 1, Eri Posta 2 1 Departet of Applied Matheatics, The Uiversity of Wester Otario, Lodo, Otario, Caada 2 Maplesoft, Waterloo wfa54@uwo.ca, deffrey@uwo.ca Abstract. We cosider soe iteger sequeces coected with cobiatorial applicatios. Specifically, we cosider Stirlig partitio ad cycle ubers, associated Stirlig partitio ad cycle ubers, ad Euleria ubers of the first ad secod ids. We cosider their evaluatio i differet cotexts. Oe cotext is the calculatio of a sigle value based o sigle iput arguets. A ore coo cotext, however, is the calculatio of a sequece of values. We copare strategies for both. Where possible, we copare with existig Maple ipleetatios. 1 Itroductio For exteded discussios of Stirlig ad Euleria ubers, we refer to [1, 2]. These ad siilar ubers arise frequetly i cobiatorial applicatios, ad have therefore bee ipleeted i several coputer algebra systes. To date, the stadard libraries of ost systes have icluded Stirlig ubers, but ot associated Stirlig ubers [3], eve though they have foud several applicatios i recet years. For exaple, they have appeared i series expasios for the Labert W fuctio [4], ad also appeared i oe for of Stirlig s series for the Gaa fuctio [2]. (Stirlig did ot defie the associated ubers.) Aother feature of ay ipleetatios is that the fuctios expect a sigle arguet, ad retur a sigle value. I practice, however, a applicatio will usually require a sequece of values, for exaple, to provide successive coefficiets i a series. The requireet of returig ultiple values has already bee recogized i soe Maple fuctios, for exaple, i the ipleetatio of Beroulli ubers: it accepts a ode paraeter. To quote fro Maple help: The ode paraeter cotrols whether or ot the beroulli routie coputes additioal Beroulli ubers i parallel with the requested oe. For exaple, if your coputer has 4 cores, the the coad beroulli(1000, sigleto=false) will copute (ad store) beroulli(1002) beroulli(1004), ad beroulli(1006). Sice i practice early all coputatios which use Beroulli ubers require ay of the, ad require the i sequece, this results i cosiderable efficiecy gais.

3 This paper addresses both the coputatio of sigle values ad of the iteger sequeces associated with the cobiatorial fuctios uder cosideratio. As a atter of teriology, we shall call a fuctio that accepts a uique arguet ad returs the correspodig uique result a sigleto fuctio, ad the correspodig operatio a sigleto coputatio. I cotrast, a fuctio acceptig a rage (explicit or iplicit) of arguets ad returig the correspodig list of values will be a sequece fuctio, ad the calculatio a sequece calculatio. 1.1 Defiitios of ubers We collect here the defiitios of all ubers cosidered. Defiitio 1. The r-associated Stirlig ubers of the first id, ore briefly Stirlig r-cycle ubers, are defied by the geeratig fuctio r 1 1 l 1 z z =! [ ] z. (1) =1 0! Rear 1. The uber [ ] gives the uber of perutatios of distict obects ito cycles, each cycle havig a iiu cardiality r [2, p 256]. Defiitio 2. The r-associated Stirlig ubers of the secod id, called ore briefly here Stirlig r-partitio ubers, are defied, usig Karaata Kuth otatio, by the geeratig fuctio r 1 e z z =! { } z. (2)! =0 0! Rear 2. The uber { } gives the uber of partitios of a set of size ito subsets, each subset havig a iiu cardiality of r [2, 5, 6]. Defiitio 3. The Euleria ubers of the first id are defied as the uber of perutatios π 1 π 2... π of {1, 2,... } that have ascets, i.e. places where π < π +1. Defiitio 4. The Euleria ubers of the secod id are defied as the uber of perutatios of the ultiset {1, 1, 2, 2,...,, } for which all ubers betwee the two occurreces of every, with 1, are greater tha, for each perutatio havig ascets, i.e. places where π < π +1. Rear 3. Note that is ot a arguet. For exaple, give the ultiset {112233}, perutatios such as or are peritted, but is ot. Aogst these peritted perutatios, we cout those with ascets. Noeclature: I [1], the ubers are called siply Euleria ubers, while the ubers are called secod-order Euleria ubers. 2

4 2 Stirlig partitio ubers The Maple 2017 ipleetatio is a sigleto fuctio, deoted stirlig2 i the cobiat pacage. It uses the forula { } = 1 ( ) ( 1). (3)! =0 For the sigleto coputatio, Table1 shows that the ties 3 are uch less usig (3). I this table, we copared the Maple fuctio stirlig2 with the ethod give below usig the recurrece relatio (7). Tiigs for a sequece calculatio, however, give i Table 2, show the ew ethod is ore efficiet. recurrece stirlig Table 1. Tiigs (sec) for geeratig a sigleto Stirlig Partitio uber. The tie usig (7) is copared with the Maple stirlig2 fuctio. 2.1 Sequece calculatio Give,, we wish to copute all Stirlig partitio ubers { i } such that i ad. We use the recurrece relatio { i } { i 1 = subect to the boudary coditios { } = 1, ad } + { i 1 1 }, (4) { } i = 1. (5) 1 Sice { i } = 0 for > i (see Fig. 1), we defie a atrix P which will ot store these zeros. { } i + 1 P i =. (6) The the recurrece relatio becoes P (i, ) = P (i 1, ) + P (i, 1). (7) The boudary coditios the becoe, respectively, P (1, ) = { } = 1, ad P (i, 1) = 1. 3 Product placeet: ties foud usig a Itel i7 i a Leovo Ultraboo. 3

5 Tiigs Table 2 shows the tiigs for fillig atrices of various sizes with iteger sequeces of Stirlig partitio ubers. The recurrece relatio (7) is copared with creatig each etry through a call to Maple s stirlig2. Fillig the square atrix P (, ) actually calculates all partitio ubers { i } with i 2 ad. This is doe for tiig coveiece, ad the atrix ca be reshaped for other applicatios. recurrece stirlig Table 2. Tiigs (sec) for geeratig sequeces of Stirlig Partitio ubers. The tie usig (7) copared with Maple stirlig2 fuctio. 3 Stirlig cycle ubers We cosider the coputatio of [ ], ipleeted i Maple 2017 as stirlig1 i the cobiat pacage. The coputatioal ethod used by stirlig1 is based o Stirlig s origial defiitio of his ubers: x = [ ] ( 1) x. (8) For give, stirlig1 costructs the product o the left, which is the collected i powers of x, so that by equatig the coefficiets of x, all ubers [ ] for 1 are deteried ad stored. Thus, a future call to [ ] with 1 will be retured by table looup, but a future call with a differet will iitiate a ew coputatio. It is iterestig that although the iterface appears to offer the user oly a sigleto coputatio, i fact a particular iteger sequece has bee coputed siletly. 3.1 Sigleto coputatio A sigleto coputatio returs the value of a fuctio for a sigle pair of iput arguets. We ipleet the ow recurrece relatio [ ] = ( 1) [ 1 4 ] + [ ] 1, (9) 1

6 subect to boudary coditios [ ] =1, for 1, (10) [ ] =( 1)!. (11) 1 We defie the vector u (i) = [ ] i + 1. I Fig. 1, we see that for fixed i, u (i) describes ubers alog the ith diagoal lie, coutig fro the left. The recurrece relatio (9) ca be writte i ters of u as u (i) = (i + 2)u (i 1) + u (i) 1, with u (i) 1 = (i 1)!. We ote that oce u (i 1) is used, it does ot eed to be stored further, so we ca overwrite storage. Our iteratio schee is thus (Maple otatio for the ith eleet of a vector is u[i]) u[] = (i + 2)u[] + u[ 1]. Therefore, we iitialize u[1] = u (i) 1 = [ i 1] = (i 1)! ad fill i diagoal lies successively. Coplexity The ai of this subsectio is to gai isight ito the best ways to test the ipleetatios, by idetifyig the worse cases for the ethods. A full bit coplexity is beyod the scope of this paper, ad will require ore wor o estiates for the sizes of Stirlig ubers. As poited out by Wilf [9], the available estiates are for [ ] whe is fixed ad, whereas the preset algoriths require owledge of the opposite case. I order to calculate the uber [ ], a vector of legth ust be recoputed (overwritte) ties. Each iteratio requires oe ultiplicatio ad 3 additios. Therefore the coplexity is ( ). We ca therefore expect that the worst case for the ethod will be = /2. Sice Maple s approach ad the preset oe calculate differet sets of ubers, a direct copariso is ot very eaigful, ad so we siply ae a brief copariso betwee oe-tie calculatios. Notice that i Table 3, the ties tae by stirlig1 are approxiately idepedet of as expected. 3.2 A fiite su For copleteess, we etio that a sigleto cycle uber ca be foud fro a fiite su, as was doe for a sigleto partitio uber. We have [ ] = =0 ( 1) + ( )( 2 ){ } +. (12) 5

7 Fig. 1. Schee for calculatig sigleto Stirlig cycle [ { ] or partitio ubers }. The coputatio proceeds fro left to right ad botto to top. At each stage oly which is progressively overwritte. The ope circles show the base of each successive loop. The blac filled circles show the recurrece relatio used. The larger circle is calculated fro the two saller oes. The triagles lie show the poits coputed by oe call to stirlig1. the ubers o oe diagoal eed to be stored i the vector u (i) stirlig1 Preset schee Table 3. Ties for a sigle call to Maple s stirlig1 ad the preset sigleto coputatio. Tiigs (sec) based o 10 trials, with eory beig cleared before each call. 6

8 Cobiig this with (3), we ca express a cycle uber as a double su. This, however, is too slow to warrat further cosideratio. 3.3 Sequece calculatio The ethod used above for partitio ubers ca be readily adapted for cycle ubers. Give,, we copute all Stirlig cycle ubers [ i ] such that i ad. We use the recurrece relatio [ ] [ ] [ ] i i 1 i 1 = (i 1) +, (13) 1 subect to the boudary coditios [ ] = 1, ad [ ] i = (i 1)!. (14) 1 Sice [ i ] = 0 for > i (see Fig. 1), we defie a atrix C which will ot store these zeros. [ ] i + 1 C i =. (15) The the boudary coditios are C(1, ) = [ ] = 1, ad C(i, 1) = (i 1)!. The recurrece relatio becoes C(i, ) = (i + 2) C(i 1, ) + C(i, 1). (16) Tiigs Table 4 shows the tiig for fillig atrices of various sizes with iteger sequeces of Stirlig cycle ubers. The recurrece relatio (16) is copared with creatig each etry through a call to Maple s stirlig1. Fillig the square atrix C(, ) actually calculated all cycle ubers [ i ] with i 2. This is doe for tiig purposes, ad the atrix ca be reshaped for other applicatios. The copariso is to copute the sae ubers usig the sequece calculatio fuctio stirlig1. Larger values of (, ) are ot tabulated because a bug i Maple 2016 (ad earlier) caused larger arguets to fail. This will be corrected i Maple Associated Stirlig ubers There are o ow aalogues of (3) or (8) for the associated Stirlig ubers for r 2; hece we ust use either the geeratig fuctios (2) ad (1), or the followig recurrece relatios. { } { } ( ){ } + 1 r + 1 = +, (17) r 1 1 [ ] + 1 [ ] = + r 1 [ r ]. (18) 7

9 recurrece stirlig Table 4. Tiigs (sec) for geeratig sequeces of Stirlig cycle ubers. The tie usig (16) copared with Maple s stirlig1 fuctio. Note that 0 = 1. The boudary cases are { } = 1, r, (19) 1 [ ] = ( 1)!, r, (20) 1 { } r = (r)! (r!)!, 1, (21) [ ] r = (r)! r!, 1. (22) 4.1 Sigleto Stirlig 2-partitio ad 2-cycle The two coputatios have the sae structure, ad ca be described i parallel. We choose to ipleet { } { } { } 1 2 = + ( 1), (23) [ ] [ ] 1 = ( 1) We also have boudary coditios We defie the vector [ ] 2 = [ ] { } 2 1 [ ] 2 + ( 1) 1 = (2)! = (2 1)!!!2 { } (2 + 1)! = 2 3( 1)!2 = 2 u (i) = [ ] i ,. (24)

10 ad siilarly for 2-partitio ubers. I Fig. 2, we see that if we fix i, the u (i) describes ubers alog the ith diagoal lie. Now u (i) 1 = i! ad u (i) = (i + 2 2)u (i 1) + (i + 2 2)u (i) 1. We ote that oce u (i 1) is used, it does ot eed to be stored further, so we ca overwrite storage. Our iteratio schee is thus u[] = (i + 2 2)(u[] + u[ 1]). For iitializatio, we ca use a special case of (24): [ ] [ ] = u (1) = (2 + 1) = (2 + 1)u (1). Therefore, we iitialize u to i = 1 usig u (1) = u[] = 1 ad fill i oe lie at a tie by fixig i ad loopig over. Each loop starts settig u (i) 1 = i! = iu (i 1) 1. We the loop over i. Fig. 2. Calculatig 2-partitio ad 2-cycle ubers. As with the r = 1 case, oly ubers o oe slopig lie eed to be ept at ay stage of the coputatio. The sae covetio for illustratig the recurrece relatio is used. 4.2 Sequece calculatio of 2-partitio ad 2-cycle ubers Give,, we copute all Stirlig 2-partitio ubers { } i or 2-cycle ubers [ i ] such that i ad. We use the recurrece relatios (23) or (24) as 9

11 appropriate. Sice { i } = [ ] i = 0 for 2 > i (see Fig. 2), we defie a atrix C which will ot store these zeros. [ ] i C i =. (25) The the recurrece relatio for 2-partitio becoes C(i, ) = C(i 1, ) + (i + 2 2)C(i 1, 1). (26) The recurrece relatio for 2-cycle becoes C(i, ) = (i + 2 2) C(i 1, ) + (i + 2 2)C(i 1, 1). (27) 4.3 Sigleto Stirlig r-partitio ad r-cycle ubers Fro the above discussio of 1-associated ad 2-associated ubers, the geeralizatio is clear. We have to ipleet [ ] [ ] [ ] + 1 r + 1 = + ( 1)( 2)... ( r + 2). (28) 1 We defie the vector u (i) = [ ] i + r 1 The geeralizatio of Fig. 2 to oe cotaiig lies of slope 1/r is ot show. For fixed i, u (i) describes ubers alog oe of the lies, with u (i) 1 = (i + r 2)! ad u (i) = (i + r 2)u (i 1). + (i + r 2)(i + r 3)... (i + r r)u (i) 1. We ote that oce u (i 1) is used, it does ot eed to be stored further, so we ca overwrite storage. Our iteratio schee is thus u[] = (i + r 2)u[] + (i + r 2)... (i + r r)u[ 1]. For iitializatio, we ca use a special case of (28): [ ] [ ] r + r = u (1) r =(r + r 1)(r + r 2)... (r + 1) =(r + r 1)(r + r 2)... (r + 1)u (1). We iitialize u usig u (1) = u[] = 1 ad fill i each lie by fixig i ad loopig over. We start each loop by settig u (i) 1 = (i + r 2)! = (i + r 2)u (i 1) 1. We the loop over i. 10

12 4.4 Sequece calculatio of r-partitio ad r-cycle ubers Give,, we copute all Stirlig r-partitio ubers { } i or r-cycle ubers [ i ] such that i ad. The recurrece relatio applied here ca refer to (17) ad (18), which is subect to the boudary coditios Sice { i } = [ ] i these zeros. { } 1 = 1 [ ] 1 = 1. (29) 1 = 0 for r > i, we defie a atrix C which will ot store [ ] i + r 1 C i =. (30) The the recurrece relatio for r-partitio becoes ( ) i + r 2 C(i, ) = C(i 1, ) + C(i r + 1, 1). (31) r 1 The recurrece relatio for r-cycle becoes C(i, ) = (i+r 2) C(i 1, )+(i+r 2)(i+r 3)... (i+r r) C(i r+1, 1) (32) 4.5 Ipleetatio i Maple I our ipleetatio of Stirlig ubers, we provide procedures for users to copute either a sigleto Stirlig uber or a sequece of Stirlig ubers. The procedures are 1. StirligRCycle: to calculate a sigleto Stirlig r-cycle uber. 2. StirligRCycleMatrix: to calculate a sequece of Stirlig r-cycle ubers. 3. StirligRPartitio: to calculate a sigleto Stirlig r-partitio uber. 4. StirligRPartitioMatrix: to calculate a sequece of Stirlig r-partitio ubers. Neither Maple or Matheatica has a ipleetatio with which to copare our progras. Therefore we have prograed the recurrece relatios, as well as the geeratig fuctios i Maple. I Table 5 below, we copared our ew schee for coputig a sigleto r-associated Stirlig cycle uber with usig the geeratig fuctio. The geeratig fuctio for Stirlig r-cycle ubers is: StirRCycleGe := proc(,,r) local t, z, p; t:=series((l(1/(1-z)) - add(z^p/p, p=1..r-1))^, z=0,+1);!*coeff(t, z, )/!; ed proc; 11

13 Sigleto schee Geeratig fuctio Table 5. Tiigs i secods of coputatios of sigle Stirlig r-cycle uber. Colu headigs give the fuctios used. The ubers tested were [ ] Table 5 shows that the sigleto schee is uch faster tha the geeratig fuctio for the coputatio of sigle r-associated Stirlig cycle uber. For the coputatio of a sequece of r-associated Stirlig cycle ubers, we copared three ethods: (1) a loop callig the sigleto fuctio; (2) a loop callig the geeratig fuctio; (3) the sequece procedure. The results are collected i Tables 6 ad 7, ad show that the sequece procedure is fastest. Sigleto schee Geeratig fuctio Table 6. Tiigs i secods of coputatios of a sequece of r-associated Stirlig cycle ubers. Colu headigs give the fuctios used. The iput arguet is, ad the retur is a atrix. Siilar tests were perfored for r-associated Stirlig partitio ubers. The geeratig fuctio for Stirlig r-partitio ubers is: StirRPartGe := proc(,,r) local t, z, p; t:=series((exp(z) - add(z^p/p!, p=0..r-1))^, z=0, +1);!*coeff(t, z, )/!; ed proc; The test data are collected i Tables 8, 9, 10. Sice the patter is siilar to that for cycle ubers, the discussio ad tables are abbreviated. 12

14 Sigleto schee Sequece schee Table 7. Tiigs i secods of coputatios of a sequece of r-associated Stirlig cycle ubers. Colu headigs give the fuctios used. The iput arguet is, ad the retur is a atrix. Sigleto schee Geeratig fuctio Table 8. Tiigs i secods of coputatios of sigle r-associated Stirlig partitio uber. Colu headigs give the fuctios beig used. The ubers tested were { 1000 } 18. Sigleto schee Geeratig fuctio Table 9. Tiigs i secods of coputatios of a sequece of r-associated Stirlig partitio uber. Colu headigs give the fuctios beig used. The iput arguet is, ad the retur is a atrix. Sigleto schee Sequece schee Table 10. Tiigs i secods of coputatios of a sequece of r-associated Stirlig partitio ubers. Colu headigs give the fuctios used. The iput arguet is, ad the retur is a atrix. 13

15 5 A ultiple threads approach to sequece calculatios The Maple help for Beroulli ubers, quoted i the itroductio, states that additioal values of Beroulli ubers are calculated i parallel. This sectio explored ways i which parallel coputatio could be applied to Stirlig ubers. For this, we use the Threads pacage i Maple. Whe we geerate the ubers iside a atrix, istead of fillig the atrix row by row ad colu by colu, we fill each diagoal fro left to right. Here is the ai part i the sequetial code to fill Stirlig r-cycle ubers i the atrix by diagoal with give iput arguets (, r) where is the size of atrix. for N fro 3 to do for fro 2 to N-1 do pd := ul(n-+r-1-l, l = 1.. r-1); A(N-+r, ) := pd*a(n-, -1)+(N-+r-2)*A(N-+r-1, ); ed do; ed do; Accordig to the recurrece relatio, we ow that we ca divide such diagoal ito a left half ad right half. So we defie two subrouties accordigly. fileft := proc (N, r) local, Nsplit, pd, l; global A; Nsplit := floor((1/2)*n+1/2); for fro 2 to Nsplit do pd := ul(n-+r-1-l, l = 1.. r-1); A(N-+r, ) := pd*a(n-, -1)+(N-+r-2)*A(N-+r-1, ); ed do; ed proc; ad filrght := proc (N, r) local, Nsplit, pd, l; global A; Nsplit := floor((1/2)*n+1/2); for fro Nsplit+1 to N-1 do pd := ul(n-+r-1-l, l = 1.. r-1); A(N-+r, ) := pd*a(n-, -1)+(N-+r-2)*A(N-+r-1, ); ed do; ed proc; Ad for each half of the diagoal, we ca establish a idepedet thread to fulfill the tas. We ipleeted this approach i Maple. Threaded := proc (, r) local N,, Nsplit; global A; A := Matrix(,, fill = 0); A(1, 1) := 1; for N fro 3 to do Threads:-Tas:-Start(ull, Tas = [fileft, N, r], Tas = [filrght, N, r]) ed do; ed proc; 14

16 Table 11 copares the threaded schee with the sequetial schee i the coputatio of a atrix of Stirlig cycle ubers. The table reflects the liitatio that there is a overhead cost to settig up ew threads, ad the beefit of the threaded approach is felt oly whe the aout of wor achieved withi a thread outweighs the overhead. I this ipleetatio, ew threads are created for each loop. We are explorig ew ethods of calculatio which will allow the threads to wor ore efficietly, with less overhead. Threaded schee Sequetial schee Table 11. Tiigs i secods of copariso of threaded code with sequetial code i geeratig sequeces of Stirlig Cycle ubers. The tests were ade o a AMD 8-core processor. 6 Ipleetatio of Euleria ubers The Euleria ubers share ay siilarities with the Stirlig ubers, ad all the ethods described above ca be applied to their case. The ubers obey the followig recurrece relatios [1]. 1 1 = ( + 1) + ( ), (33) 1 = ( + 1) 1 + (2 1) 1 1. (34) The preset Maple fuctios euleria1 ad euleria2 are recursively prograed ipleetatios of these equatios. As a cosequece, they are very slow for large arguets. The ew ipleetatio of these ubers cosists of 4 fuctios, which follow the patters of the Stirlig uber ipleetatios. 1. Euleria1: calculates a sigleto Euleria uber of the first id. As with Stirlig partitio ubers, a fiite su is ow which is distictly the fastest ethod for a sigleto coputatio [8]: +1 ( ) + 1 = ( 1) ( + 1). (35) =0 15

17 2. Euleria1Matrix: calculates a sequece of Euleria ubers of the first id. This follows the sequece calculatio of Stirlig ubers, usig (33). 3. Euleria2: calculates sigleto Euleria ubers of the secod id. This follows the sipleto ethod used earlier for Stirlig cycle ubers. 4. Euleria2Matrix: calculates a sequece of Euleria ubers of the secod id. This follows the sequece calculatio of Stirlig ubers. 6.1 Tiigs for Euleria uber calculatios I view of the siilarities with Stirlig ubers, we shall ot labour the coparisos betwee ethods, sice they for the sae processio of speeds see before. Table 12 copares the ew ipleetatios, followig the patters set above. Euleria1 Euleria1Matrix Euleria2 Euleria2Matrix Table 12. Tiigs i secods of coputatios of Euleria ubers. Colu headigs give the fuctios used. Refereces 1. Graha, R.L., Kuth, D.E., Patashi, O., Cocrete Matheatics, Addiso- Wesley Publishig Co., Readig, Massachusetts, Cotet, L., Advaced Cobiatorics, D. Reidel Publishig Co., Dordrecht, Hollad, Howard, F. T., Associated Stirlig Nubers, The Fiboacci Quarterly, Vol. 18(4), , Corless, R.M., Jeffrey, D.J., Kuth, D.E., A Sequece of Series for the Labert W Fuctio, Proceedigs of ISSAC 1997, ed. W.W. Kuechli, ACM Press, Karaata, J., Theoree sur la soabilite expoetielle et d autres soabilites rattachat, Matheatica, Clu, Roaia, vol. 9, , Kuth, D.E., Two Notes o Notatio, The Aerica Matheatical Mothly, vol. 99, , Stirlig, J., Methodus Differetialis, Lodo, Leher, D. H., Geeralized Euleria Nubers, Joural of Cobiatorial Theory, Series A 32, , Wilf, H. S., The Asyptotic Behavior of the Stirlig Nubers of the First Kid, Joural of Cobiatorial Theory, Series A 64, ,