Research Article Zeons, Permanents, the Johnson Scheme, and Generalized Derangements

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1 Hindawi Publishing Corporaion Inernaional Journal of Combinaorics Volume 211, Aricle ID 5393, 29 pages doi:1.1155/211/5393 Research Aricle Zeons, Permanens, he Johnson Scheme, and Generalized Derangemens Philip Feinsilver and John McSorley Deparmen of Mahemaics, Souhern Illinois Universiy, Carbondale, IL 6291, USA Correspondence should be addressed o Philip Feinsilver, pfeinsil@mah.siu.edu Received 2 January 211; Acceped 1 April 211 Academic Edior: Alois Panholzer Copyrigh q 211 P. Feinsilver and J. McSorley. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Saring wih he zero-square zeon algebra, he connecion wih permanens is shown. Permanens of submarices of a linear combinaion of he ideniy marix and all-ones marix lead o momen polynomials wih respec o he exponenial disribuion. A permanen race formula analogous o MacMahon s maser heorem is presened and applied. Connecions wih permuaion groups acing on ses and he Johnson associaion scheme arise. The families of numbers appearing as marix enries urn ou o be relaed o ineresing variaions on derangemens. These generalized derangemens are considered in deail as an illusraion of he heory. 1. Inroducion Funcions acing on a finie se can be convenienly expressed using marices, whereby he composiion of funcions corresponds o muliplicaion of he marices. Essenially, one is considering he induced acion on he vecor space wih he elemens of he se acing as a basis. This acion exends o ensor powers of he vecor space. One can ake symmeric powers, anisymmeric powers, and so forh, ha yield represenaions of he muliplicaive semigroup of funcions. An especially ineresing represenaion occurs by aking nonreflexive, symmeric powers. Idenifying he underlying se of cardinaliy n wih {1, 2,...,n}, he vecor space has basis e 1,e 2,... The acion we are ineresed in may be found by saying ha he elemens e i generae a zeon algebra, he relaions being ha he e i commue, wih e 2 i, 1 i n. To ge a feeling for his, firs we recall he acion on Grassmann algebra where he marix elemens of he induced acion arise as deerminans. For he zeon case, permanens appear. An ineresing connecion wih he cenralizer algebra of he acion of he symmeric group comes up. For he defining acion on he se {1,...,n}, represened as -1 permuaion

2 2 Inernaional Journal of Combinaorics marices, he cenralizer algebra of n n marices commuing wih he enire group is generaed by I, he ideniy marix, and J, he all-ones marix. The quesion was if hey would help deermine he cenralizer algebra for he acion on subses of a fixed size, l-ses, for l > 1. I is known ha he basis for he cenralizer algebra is given by he adjacency marices of he Johnson scheme. Could one find his working solely wih I and J? The resul is ha by compuing he zeon powers, ha is, he acion of si J, linear combinaions of I and J, on l-ses, he Johnson scheme appears naurally. The coefficiens are polynomials in s and occurring as momens of he exponenial disribuion. And hey urn ou o coun derangemens and relaed generalized derangemens. The occurrence of Laguerre polynomials in he combinaorics of derangemens is well known. Here, he 2 F 1 hypergeomeric funcion, which is closely relaed o Poisson-Charlier polynomials, arises raher naurally. Here is an ouline of he paper. Secion 2 inroduces zeons and permanens. The race formula is proved. Connecions wih he cenralizer algebra of he acion of he symmeric group on ses are deailed. Secion 3 is a sudy of exponenial polynomials needed for he remainder of he paper. Zeon powers of si J are found in Secion 4 where he specra of he marices are found via he Johnson scheme. Secion 5 presens a combinaorial approach o he zeon powers of si J, including an inerpreaion of exponenial momen polynomials by elemenary subgraphs. In Secion 6, generalized derangemen numbers, specifically couning derangemens and couning arrangemens, are considered in deail. The Appendix has some derangemen numbers and arrangemen numbers for reference, as well as a page of exponenial polynomials. An example expressing exponenial polynomials in erms of elemenary subgraphs is given here. 2. Represenaions of Funcions Acing on Ses Le V denoe he vecor space Q n or R n. We will look a he acion of a linear map on V exended o quoiens of ensor powers V l. We work wih coordinaes raher han vecors. Firs, recall he Grassmann case. To find he acion on V l consider an algebra generaed by n variables e i saisfying e i e j e j e i. In paricular, e 2 i. Noaion. The sandard n-se {1,...,n} will be denoed n. Roman caps I, J, A, and so forh denoe subses of n. We will idenify hem wih he corresponding ordered uples. Generally, given an n-uple x 1,...,x n and a subse I n, we denoe producs x I j I x j, 2.1 where he indices are in increasing order if he variables are no assumed o commue. As an index, we will use U o denoe he full se n. Ialic I and J will denoe he ideniy marix and all-ones marix, respecively. For a marix X IJ, say, where he labels are subses of fixed size l, dicionary ordering is used. Tha is, conver o ordered uples and use dicionary ordering. For example, for n 4, l 2, we have labels 12, 13, 14, 23, 24, and 34 for rows one hrough six, respecively.

3 Inernaional Journal of Combinaorics 3 AbasisforV l is given by producs e I e i1 e i2 e il, 2.2 I n, where we consider I as an ordered l-uple. Given a marix X acing on V,le y i j X ij e j, 2.3 wih corresponding producs y I, hen he marix X l has enries given by he coefficiens in he expansion y I J ( X l) IJe J 2.4 where he anicommuaion rules are used o order he facors in e J. Noe ha he coefficien of e j in y i is X ij iself. And for n>3, he coefficien of e 34 in y 12 is X13 X 14 de. 2.5 X 23 X 24 We see ha in general he IJ enry of X l is he minor of X wih row labels I and column labels J. A sandard erm for he marix X l is a compound marix. NoinghaX l is n l n l, in paricular, l n yields he one-by-one marix wih enry equal o X n UU de X. In his work, we will use he algebra of zeons, sanding for zero-ons, or more specifically, zero-square bosons. Tha is, we assume ha he variables e i saisfy he properies e i e j e j e i, e 2 i. 2.6 A basis for he algebra is again given by e I,I n. A level l, he induced marix X l has IJ enries according o he expansion of y I, y i j X ij e j y I J ( X l) IJe J 2.7 similar o he Grassmann case. Since he variables commue, we see ha he IJ enry of X l is he permanen of he submarix wih rows I and columns J. In paricular, X n UU per X.We refer o he marix X l as he lh zeon power of X Funcions on he Power Se of n Noe ha X l is indexed by l-ses. Suppose ha X f represens a funcion f : n n. So i is a zero-one marix wih X f ij 1, he single enry in row i if f maps i o j.

4 4 Inernaional Journal of Combinaorics I J B A l j C B C k j k k Figure 1: Configuraion of ses: B I \ A, C J \ A. The lh zeon power of X is he marix of he induced map on l-ses. If f maps an l-se I o one of lower cardinaliy, hen he corresponding row in X l has all zero enries. Thus, he induced marices in general correspond o parial funcions. However, if X is a permuaion marix, hen X l is a permuaion marix for all l n. So, given a group of permuaion marices, he map X X l is a represenaion of he group Zeon Powers of si X Our main heorem compues he lh zeon power of si X for an n n marix X, where s and are scalar variables. Figure 1 illusraes he proof. Theorem 2.1. For a given marix X, for l n, and indices I J l, ( si X l) s l j j IJ j l A I J A l j ( X j) I\A,J\A. 2.8 Proof. Sar wih y i se i ξ i, where ξ i j X ije j.giveni i 1,...,i l, we wan he coefficien of e J in he expansion of he produc y I y i1 y il.now, y I se i1 ξ i1 se il ξ il. 2.9 Choose A Iwih A l j, j l. A ypical erm of he produc has he form s l j j e A ξ B 2.1 where A B, B I \ A. ξ B denoes he produc of erms ξ i wih indices in B. Expanding, we have ξ B C ( X j) BC e C, e A ξ B C ( X j) BC e Ae C. 2.11

5 Inernaional Journal of Combinaorics 5 Thus, for a conribuion o he coefficien of e J, we have A C J, where A C. Thais, C J \ AandA I J. So, he coefficien of s l j j is as saed Trace Formula Anoher main feaure is he race formula which shows he permanen of I X as he generaing funcion for he races of he zeon powers of X. This is he zeon analog of he heorem of MacMahon for represenaions on symmeric ensors. Theorem 2.2. One has he formula per si X j s n j j r X j Proof. The permanen of si X is he UU enry of si X n. Specialize I J Uin Theorem 2.1.SoAisany n j -se wih I \ A J \ A A, is complemen in n.thus, per si X ( si X n) UU ( s n j j X j) A,A j n A n j s n j j r X j, j n 2.13 as required Permuaion Groups Le X be an n n permuaion marix. We can express per I X in erms of he cycle decomposiion of he associaed permuaion. Proposiion 2.3. For a permuaion marix X, per I X l n ( 1 l) n X l, 2.14 where n X l is he number of cycles of lengh l in he cycle decomposiion of he corresponding permuaion. Proof. Decomposing he permuaion associaed o X yields a decomposiion ino invarian subspaces of he underlying vecor space V. So per I X will be he produc of per I X c as c runs hrough he corresponding cycles wih X c he resricion of X o he invarian subspace for each c. So we have o check ha if X acs on V l as a cycle of lengh l, hen per I X 1 l. For his, apply Theorem 2.2. Apar from level zero, here is only one se fixed by any X j, namely when j l. So he race of X j is zero unless j l and hen i is one. The resul follows.

6 6 Inernaional Journal of Combinaorics Cycle Index: Orbis on l-ses Now, consider a group, G, of permuaion marices. We have he cycle index Z G z 1,z 2,...,z n 1 G z n X 1 1 z n X 2 2 z n X n X G n, 2.15 each z l corresponding o l-cycles in he cycle decomposiion associaed o he Xs. From Proposiion 2.3, we have an expression in erms of permanens. Combining wih he race formula, we ge he following. Theorem 2.4. Le G be a permuaion group of marices, hen one has 1 G ( per I X Z G 1, 1 2,...,1 n) X G l l # orbis on l-ses Remark 2.5. This resul refers o hree essenial heorems in group heory acing on ses. Equaliy of he firs and las expressions is he permanen analog of Molien s heorem, which is he case for a group acing on he symmeric ensor algebra, ha he cycle index couns orbis on subses is an insance of Polya Couning, wih wo colors. The las expression is followed by he Cauchy-Burnside lemma applied o he groups G l {X l } X G Cenralizer Algebra and Johnson Scheme Given a group, G, of permuaion marices, an imporan quesion is o deermine he se among all marices of marices commuing wih all of he marices in G.Thisishecenralizer algebra of he group. For he symmeric group, he only marices are I and J. For he acion of he symmeric group on l-ses, a basis for he cenralizer algebra is given by he incidence marices for he Johnson disance. These are he same as he adjacency marices for he Johnson associaion scheme. Recall ha he Johnson disance beween wo l-ses I and J is dis JS I, J 1 2 I Δ J I \ J J \ I The corresponding marices JS nl k ( are defined by JS nl k ) IJ 1 if dis JS I, J k,, oherwise As i is known, 1, page 36, ha a basis for he cenralizer algebra is given by he orbis of he group G 2, acing on pairs, he Johnson basis is a basis for he cenralizer algebra. Since he Johnson disance is symmeric, i suffices o look a G 2.

7 Inernaional Journal of Combinaorics 7 Now, we come o he quesion ha is a saring poin for his work. If I and J are he only marices commuing wih all elemens as marices of he symmeric group, hen since he map G G l is a homomorphism, we know ha I l and J l are in he cenralizer algebra of G l. The quesion is how o obain he res? The, perhaps surprising, answer is ha in fac one can obain he complee Johnson basis from I and J alone. This will be one of he main resuls, Theorem Permanen of si J Firs, le us consider si J. Proposiion 2.6. One has he formula per si J n! s l n l l! l n Proof. For X J, we see direcly, since all enries equal one in all submarices, ha ( J l) l! 2.2 IJ for all I and J. Taking races, n r J l l l!, 2.21 and by he race formula, Theorem 2.2, per si J l n l! s n l l l l n! n l! sn l l Reversing he order of summaion yields he resul saed. Corollary 2.7. For varying n, one will explicily denoe p n s, per si n J n, hen, wih p s, 1, n The Corollary exhibis he operaional formula z n n! p n s, esz 1 z p n s, where D s d/ds. By inspecion, his agrees wih 2.19 as well. 1 1 D s s n, 2.24

8 8 Inernaional Journal of Combinaorics Observe ha 2.19 can be rewrien as per si J ( s y ) ne y dy, 2.25 ha is, hese are momen polynomials for he exponenial disribuion wih an addiional scale parameer. We proceed o examine hese momen polynomials in deail. 3. Exponenial Polynomials For he exponenial disribuion, wih densiy e y defined as on,, hemomen polynomials are h n x ( x y ) ne y dy. 3.1 The exponenial embeds naurally ino he family of weighs of he form x m e x on, as for generalized Laguerre polynomials. We define correspondingly h n,m x, ( x y ) n ( y ) me y dy, 3.2 for nonnegaive inegers n, m, inroducing a facor of y m and a scale facor of. We refer o hese as exponenial momen polynomials. Proposiion 3.1. Observe he following properies of he exponenial momen polynomials. 1 The generaing funcion 1 m m! n z n n! h n,m x, e zx 1 z 1 m, 3.3 for z < 1. 2 The operaional formula 1 m m! h n,m x, I D m 1 x n, 3.4 where I is he ideniy operaor and D d/dx. 3 The explici form h n,m x, n n (m ) j!x n j m j. 3.5 j j

9 Inernaional Journal of Combinaorics 9 Proof. For he firs formula, muliply he inegral by z n /n! and sum o ge y m e zx zy y dy e zx y m e y 1 z dy, 3.6 which yields he saed resul. For he second, wrie m m! I D m 1 x n m y m e I D y x n dy ( y ) me y ( x y ) n dy, 3.7 using he shif formula e ad f x f x a. For he hird, expand x y n by he binomial heorem and inegrae. A variaion we will encouner in he following is n m n m (m ) h n m,m x, j!x n m j m j 3.8 j j n n m j!x n j j j m j m n n m j!x n j j n j j m n m n m (n ) j!x j n j, 3.11 j j replacing he index j j m for 3.9 and reversing he order of summaion for he las line. And for fuure reference, consider he inegral formula h n m,m x, ( x y ) n m ( y ) me y dy Hypergeomeric Form Generalized hypergeomeric funcions provide expressions for he exponenial momen polynomials ha are ofen convenien. In he presen conex, we will use 2 F funcions, defined by 2F ( a, b ) x a j b j x j, j! j 3.13 where a j Γ a j /Γ a is he usual Pochhammer symbol. In paricular, if a, for example, is a negaive ineger, he series reduces o a polynomial. Rearranging facors in he

10 1 Inernaional Journal of Combinaorics expressions for h n,m,via 3 in Proposiion 3.1, andh n m,m, 3.8, we can formulae hese as 2F hypergeomeric funcions. Proposiion 3.2. One has he following expressions for exponenial momen polynomials: ( n, 1 m h n,m x, x n m m! 2 F ), x ( m n, 1 m h n m,m x, x n m m m! 2 F ). x Zeon Powers of si J We wan o calculae si J l,hais,he n l n l marix wih rows and columns labelled by l-subses I, J {1,...,n} wih he IJ enry equal o he permanen of he corresponding submarix of si J. This is equivalen o he induced acion of he original marix si J on he lh zeon space V l. Theorem 4.1. The lh zeon power of si J is given by si J l k l l k j!s l j j JS nl k l j k j k h l k,k s, JS nl k, 4.1 where he h s are exponenial momen polynomials. Proof. Choose I and J wih I J l. ByTheorem 2.1, we have, using he fac ha all of he enries of J j are equal o j!, ( si J l) s l j j IJ j l j l A I J A l j s l j j j!. A I J A l j ( J j) I\A,J\A 4.2 Now, if dis JS I, J k, hen I J l k, and here are condiions of he sum. Hence he resul. l k l j subses A of I J saisfying he Noe ha he specializaion l n, k, recovers We can wrie he above expansion using he hypergeomeric form of he exponenial momen polynomials, Proposiion 3.2, si J l k s l k k k! 2 F ( k l, 1 k s ) JS nl k. 4.3

11 Inernaional Journal of Combinaorics Specrum of he Johnson Marices Recall, for example, 2, page 22, ha he specrum of he Johnson marices for given n and l is he se of numbers Λ nl k α i l α n l α i l i 1 k i, i i k i 4.4 where he eigenvalue for given α has mulipliciy n α n α 1. For l-ses, he Johnson disance akes values from o min l, n l, wihα aking values from ha same range The Specrum of si J l Recall ha as he Johnson marices are symmeric and generae a commuaive algebra, hey are simulaneously diagonalizable by an orhogonal ransformaion of he underlying vecor space. Diagonalizing he equaion in Theorem 4.1, we see ha he specrum of si J l is given by h l k,k s, Λ nl k α. k 4.5 Proposiion 4.2. The specrum of si J l is given by s α n l α n l α! h l α,n l α s, i l α n l α i s l i i i!, i i 4.6 for α min l, n l, wih respecive mulipliciies n α n α 1. Proof. In he sum over i in 4.4, only he las wo facors involve k. We have l i h l k,k s, 1 k i k k i ) l k k 1 s y y k i( l i e y dy seing k i m k k i ) l i m i m 1 s y y m( l i e y dy m m 4.7 s l i i i!, ( s y y ) l i ( y ) ie y dy

12 12 Inernaional Journal of Combinaorics using he binomial heorem o sum ou m. Filling in he addiional facors yields h l k,k s, Λ nl k α l α n l α i s l i i i!, 4.8 i i i k Taking ou a denominaor facor of n l α! and muliplying by s α n l α gives l α s l α i n l α i n l α i!, i i 4.9 which is precisely h l α,n l α as in he hird saemen of Proposiion 3.1. As in Proposiion 3.2, we can express he eigenvalues as follows. Corollary 4.3. The specrum of si J l consiss of he eigenvalues s l 2F ( α l, 1 n l α s ), 4.1 for α min l, n l, wih corresponding mulipliciies as indicaed above Row Sums and Trace Ideniy For he row sums, we know ha he all-ones vecor is a common eigenvecor of he Johnson basis corresponding o α. These are he valencies Λ k. For he Johnson scheme, we have l n l Λ nl k k k for example, see 2, page 219, which can be checked direcly from he formula for Λ nl k α, 4.4,wihα se o zero. Seing α inproposiion 4.2 gives 1 n l n l! h l,n l s, i l n l i i!s l i i, i i 4.12 for he row sums of si J l Trace Ideniy Terms on he diagonal are he coefficien of JS nl, which is he ideniy marix. So, he race is n n l r si J l h l, s, k!s l k k. l l k k 4.13

13 Inernaional Journal of Combinaorics 13 Cancelling facorials and reversing he order of summaion on k yields he following formula. Now, Proposiion 4.2 gives he race r si J l n! s k l k n l! k! k l r si J l α min l,n l [ ] n n l α n l α i s l i i i!. α α 1 i i i 4.15 Proposiion 4.4. Equaing he above expressions for he race yields he ideniy α min l,n l [ ] n n l α n l α i s l i i n! s j l j i!. α α 1 i i n l! j! i j l 4.16 Example 4.5. For n 4, l 2, we have s 2 2s 2 2 s 2 2 s 2 2 s 2 2 s s 2 2 s 2 2s 2 2 s 2 2 s s 2 2 s 2 2 s 2 2 s 2 2s s 2 2 s 2 2. s 2 2 s s 2 2s 2 2 s 2 2 s 2 2 s s 2 2 s 2 2 s 2 2s 2 2 s s 2 2 s 2 2 s 2 2 s 2 2 s 2 2s One can check ha he enries are in agreemen wih Theorem 4.1. The race is 6s 2 12s The specrum is eigenvalue s 2 6s 12 2, wih mulipliciy 1, eigenvalue s 2 2s, wih mulipliciy 3, 4.18 eigenvalue s 2, wih mulipliciy 2, and he race can be verified from hese as well. Remark 4.6. Wha is ineresing is ha hese marices have polynomial enries wih all eigenvalues polynomials as well, and furhermore, he exac same se of polynomials produces he eigenvalues as well as he enries. Specializing s and o inegers, a similar saemen holds. All of hese marices will have ineger enries wih ineger eigenvalues, all of which belong o closely relaed families of numbers. We will examine ineresing cases of his phenomenon laer on in his paper.

14 14 Inernaional Journal of Combinaorics 5. Permanens from si J Here, we presen a proof via recursion of he subpermanens of si J, hereby recovering Theorem 4.1 from a differen perspecive. Remark 5.1. For he remainder of his paper, we will work wih an n n marix corresponding o an l l submarix of he above discussion. Here, we have blown up he submarix o full size as he objec of consideraion. Le M n,l denoe he n n marix wih n l enries equal o s on he main diagonal, and s elsewhere. Noe ha M n, si J and M n,n J, where I and J are n n. Define P n,l per M n,l 5.1 o be he permanen of M n,l. For l, define P, 1, and, recalling 2.19, P n, per si J n j n! j! sj n j n n! s n j j. 5.2 n j! j We have also P n,n per J n! n for J of order n n. These agree a P, 1. Theorem 5.2. For n 1, 1 l n, one has he recurrence P n,l P n,l 1 sp n 1,l Proof. We have l n so n l 1 n l 1 1, ha is, he marix M n,l 1 conains a leas 1 enry on is main diagonal equal o s. Wrie he block form M n,l 1 [ s A A T M n 1,l 1 ], 5.4 wih A,,..., he 1 n 1 row vecor of all s, and A T is is ranspose. Now, compue he permanen of M n,l 1 expanding along he firs row. We ge ) P n,l 1 per M n,l 1 s per M n 1,l 1 F (A, A T,M n 1,l 1, 5.5

15 Inernaional Journal of Combinaorics 15 where F A, A T,M n 1,l 1 is he conribuion o P n,l 1 involving A. Now, ([ ]) ) A per M n 1,l 1 F (A, A T,M n 1,l 1 per A T M n 1,l 1 ([ A T ]) M n 1,l 1 per A ([ Mn 1,l 1 A T ]) per A 5.6 P n,l. Thus, from 5.5, ) P n,l 1 s per M n 1,l 1 per M n 1,l 1 F (A, A T,M n 1,l 1 sp n 1,l 1 P n,l, 5.7 and hence he resul. We arrange he polynomials P n,l in a riangle, wih he columns labelled by l and rows by n, saring wih P, 1 a he op verex P, P 1, P 1,1 P 2, P 2,1 P 2, P n 1, P n 1,n 2 P n 1,n 1 P n, P n,1 P n,n 1 P n,n The recurrence says ha o ge he n, l enry, you combine elemens in column l 1inrows n and n 1, forming an L-shape. Thus, given he firs column {P n, } n, he able can be generaed in full. Now, we check ha hese are indeed our exponenial momen polynomials. Addiionally, we derive an expression for P n,l in erms of he iniial sequence P n,. For clariy, we will explicily denoe he dependence of P n,l on s,. Theorem 5.3. For l, one has 1 he permanen of he n n marix wih n l enries on he diagonal equal o s and all oher enries equal o is P n,l s, h n l,l s, n n l j!s n j j, n j j l 5.9

16 16 Inernaional Journal of Combinaorics 2 P n,l s, l l 1 j s j P n j, s,, j j he complemenary sum is s n n n 1 l P n,l s,. l j 5.11 Proof. The iniial sequence P n, h n, as noed in 5.2. We check ha h n l,l saisfies recurrence 5.3. Saring from he inegral represenaion for h n l 1,l 1, 3.2, we have h n l 1,l 1 ( s y ) n l 1 ( y ) l 1e y dy ( s y )( s y ) n l ( y ) l 1e y dy sh n l,l 1 h n l,l, 5.12 as required, where we now idenify h n l 1,l 1 P n,l 1, h n l,l 1 P n 1,l 1,andh n l,l P n,l. And 3.1 gives an explici form for P n,l. For 2, saring wih he inegral represenaion for P n, h n,,wege l l 1 j s j j j n je s y y dy l l 1 j s j n l l je s y s y y dy j j h n l,l, ( s y ) n l ( s y s ) le y dy 5.13 as required. The proof for 3 is similar, using 3.12, P n,l h n l,l ( s y ) n l ( y ) le y dy, 5.14 and he binomial heorem for he sum si J l Revisied Now, we have an alernaive proof of Theorem 4.1.

17 Inernaional Journal of Combinaorics 17 Lemma 5.4. Le I and J be l-subses of n wih dis JS I, J k, hen per si J IJ P l,k s, Proof. Now, I J l k, so he submarix si J IJ is permuaionally equivalen o he l l marix wih l k enries s on is main diagonal and s elsewhere, ha is, o he marix M l,k. Hence, by definiion of P l,k s,, 5.1, we have he resul. Thus, he expansion in he Johnson basis is si J l k h l k,k s, JS nl k Proof. Le I and J be l-subses of n wih Johnson disance k. By definiion, he IJ enry of he LHS of 5.16 equals he permanen of he submarix from rows I and columns J, per si J IJ P l,k s, h l k,k s,,bylemma 5.4 and Theorem Now,onheRHSof 5.16, ifdis JS I, J k, he only nonzero conribuion comes from he JS nl k erm. This yields h l k,k s, 1 h l k,k s, as required Elemenary Subgraphs and Permanens There is an approach o permanens of si J via elemenary subgraphs, based on ha of Biggs 3 for deerminans. An elemenary subgraph see 3, page 44 of a graph G is a spanning subgraph of G all of whose componens are, 1, or 2 regular, ha is, all of whose componens are isolaed verices, isolaed edges, or cycles of lengh j 3. Le K l n be a copy of he complee graph K n wih verex se n in which he firs n l verices n l {1, 2,...,n l} are disinguished. We may now consider he marix M n,l as he weighed adjacency marix of K l n in which he weighs of he disinguished verices are s, wih all undisinguished verices and all edges assigned a weigh of. Le E be an elemenary subgraph of K l n, hen we describe E as having d E disinguished isolaed verices and c E cycles. The weigh of E, w E, is defined as w E s d E n d E, 5.17 a homogeneous polynomial of degree n. This leads o an inerpreaion/derivaion of P n,l s, as he permanen per M n,l. Theorem 5.5. One has he expansion in elemenary subgraphs P n,l s, 2 c E w E E Proof. Assign weighs o he componens of E as follows: each disinguished isolaed verex will have weigh s ; each undisinguished isolaed verex will have weigh ;

18 18 Inernaional Journal of Combinaorics each isolaed edge will have weigh 2 ; and each j-cycle, j 3, will have weigh j. To obain w E in agreemen wih 5.17, we form he produc of hese weighs over all componens in E. The proof hen follows along he lines of Proposiion 7.2 of 3, page 44, slighly modified o incorporae isolaed verices and wih deerminan, de, replaced by permanen, per, ignoring he minus signs. Effecively, each erm in he permanen expansion hus corresponds o a weighed elemenary subgraph E of he weighed K l n. See Figure 2 for an example wih n Associaed Polynomials and Some Asympoics Thinking of s and as parameers, we define he associaed polynomials Q n x n n x l P n,l. l l 5.19 As in he proof of 3 above, using he inegral formula 3.12, we have Q n x ( s y xy ) n e y dy n s j 1 x n j n j( n j )! j j 5.2 n! j s j 1 x n j n j. j! Comparing wih 5.2, wehavehefollowing. Proposiion 5.6. Consider Q n x n n x l P n,l s, P n, s, x. l l 5.21 And one has he following. Proposiion 5.7. As n,forx / 1, Q n x n 1 x n n!e s/ x, 5.22

19 Inernaional Journal of Combinaorics 19 l = s + s + s + P 3, = s 3 + 3s 2 + 6s s + s + s+ (s + ) 3 (s + ) 2 (s + ) 2 (s + ) 2 3 l = 1 s + s + s + s + P 3,1 = s 2 + 4s (s + ) 2 3 (s + ) 2 (s + ) 2 3 l = 2 s + s + P 3,2 = 2s (s + ) 2 3 (s + ) l = 3 P 3,3 = Figure 2: M 3,l, K l 3, P 3,l, and he 5 weighed elemenary subgraphs of K l 3 for l, 1, 2, 3. Disinguished verices are shown in bold. wih he special cases Q n 1 s n, Q n P n, n n!e s/, Q n 1 n P n,l 2 n n!e s/ 2. l l 5.23

20 2 Inernaional Journal of Combinaorics Proof. From 5.2, Q n x n! j s j 1 x n j n j j! n n 1 x n 1 s/ j n!, j! 1 x j 5.24 from which he resul follows. 6. Generalized Derangemen Numbers The formula 2.19 is suggesive of he derangemen numbers see, e.g., 4, page 18, n 1 j d n n!. 6.1 j! j This leads o he following. Definiion 6.1. A family of numbers, depending on n and l, arising as he values of P n,l s, when s and are assigned fixed ineger values, are called generalized derangemen numbers. We have seen ha he assignmen s 1, 1 produces he usual derangemen numbers when l. In his secion, we will examine in deail he cases s 1, 1, generalized derangemens,ands 1, generalized arrangemens. Remark 6.2. Topics relaed o his maerial are discussed in Riordan 5. The paper 6 is of relaed ineres as well Generalized Derangemens of n To sar, define D n,l P n,l 1, Equaion 5.9 and Proposiion 3.2 give D n,l n j l 1 n j ( n l n j ) j! 1 n l l! 2 F ( l n, 1 l ) Equaion 5.2 reads as per J I D n, d n, 6.4 he number derangemens of n. So we have a combinaorial inerpreaion of D n,.

21 Inernaional Journal of Combinaorics Combinaorial Inerpreaion of D n,l We now give a combinaorial inerpreaion of D n,l for l 1. When l 1, recurrence 5.3 for P n,l 1, 1 gives D n,l D n,l 1 D n 1,l We say ha a subse I of n is deranged by a permuaion if no poin of I is fixed by he permuaion. Proposiion 6.3. D n, d n, he number of derangemens of n. In general, for l, D n,l is he number of permuaions of n in which he se {1, 2,...,n l} is deranged, wih no resricions on he l-se {n l 1,...,n}. Proof. For l, le D denoe he se of permuaions in he saemen of he proposiion. n,l Le E n,l D n,l. We claim ha E n,l D n,l. The case l is immediae. We show ha E n,l saisfies recurrence 6.5. Now, le l>. Consider a permuaion in D.Thepoinn is eiher 1 deranged, or n,l 2 no deranged i.e., fixed. 1 If n is deranged, hen he n l 1 -se {1, 2,...,n l, n} is deranged. By swiching n n l 1 in all permuaions of D n,l, we obain a permuaion in D n,l 1. Conversely, given any permuaion of D,weswichn n l 1oobain n,l 1 a permuaion in D where n is deranged. Hence, he number of permuaions in n,l D n,l wih n deranged equals E n,l 1. 2 Here, n is fixed, so if we remove n from any permuaion in D we obain n,l a permuaion in D n 1,l 1. Conversely, given a permuaion in D, we may n 1,l 1 include n as a fixed poin o obain a permuaion in D wih n fixed. Hence, he n,l number of permuaions in D n,l wih n fixed equals E n 1,l 1. Combining he above wo paragraphs shows ha E n,l saisfies recurrence 6.5. And a quick check, D n,n n!, 6.6 here being no resricions a all in he combinaorial inerpreaion, in agreemen wih 6.3 for l n. Example 6.4. When n 3, we have d 3 D 3, 2 corresponding o he 2 permuaions of 1 in which {1, 2, 3} is moved: 231, 312. Then, D 3,1 3 corresponding o he 3 permuaions of 1 in which {1, 2} is moved: 213, 231, 312. Then, D 3,2 4 corresponding o he 4 permuaions of 1 in which {1} is moved: 213, 231, 312, 321. Finally, D 3,3 3! 6 corresponding o he 3 permuaions of 1 in which is moved: 123, 132, 213, 231, 312, 321.

22 22 Inernaional Journal of Combinaorics Reversing he order of summaion in 6.3 gives an alernaive expression n l n l (n D n,l 1 j ) j!. 6.7 j j Remark 6.5. Formulaion 6.7 may be proved direcly by inclusion-exclusion on permuaions fixing given poins. Example 6.6. Consider ) 3 D 5,2 j 1 j( 3 (5 ) j! 5! 4! 3! 2! j Now, from 2 of Theorem 5.3, s 1, and 1, we have D n,l l l d n j. j j 6.9 Here is a combinaorial explanaion. To obain a permuaion in D, we firs choose j poins n,l from {n l 1,...,n} o be fixed. Then, every derangemen of he remaining n j poins will produce a permuaion in D n,l, and here are d n j of such derangemens. Example 6.7. Consider D 5, d 5 j d 5 d 4 j 1 2 j d Permanens from J I Theorem 4.1 specializes o J I l min l,n l k D l,k JS nl k This can be wrien using he hypergeomeric form J I l min l,n l k 1 l k k! 2 F ( k l, 1 k ) 1 JS nl k, 6.12

23 Inernaional Journal of Combinaorics 23 wih specrum α l, α n l 1 eigenvalue 1 l 2F 1, n n occurring wih mulipliciy, α α by Corollary 4.3 and Proposiion 4.2. The enries of J I l are from he se of numbers D n,l. For he specrum, sar wih α. From 6.3, we have 1 l 2F ( l, n l 1 ) 1 1 n l! D n,n l As α increases, we see ha he specrum consiss of he numbers 1 α n l α! D n 2α,n l α Think of moving in he derangemen riangle, as in he appendix, saring from posiion n, n l, rescaling he values by he facorial of he column a each sep, hen he eigenvalues are found by successive knigh s moves, up 2 rows and one column o he lef, wih alernaing signs. Example 6.8. For n 5, l 3, we have J I , wih characerisic polynomial λ 5 λ 32 λ

24 24 Inernaional Journal of Combinaorics Remark 6.9. Excep for l 2, he coefficiens in he expansion of J I l in he Johnson basis will be disinc. Thus, he Johnson basis iself can be read off direcly from J I l.inhis sense, he cenralizer algebra of he acion of he symmeric group on l-ses is deermined by knowledge of he acion of jus J I on l-ses Generalized Arrangemens of n Given n, j n, aj-arrangemen of n is a permuaion of a j-subse of n. The number of j-arrangemens of n is A ( n, j ) n! ( n j )! Noe ha here is a single -arrangemen of n, from he empy se. Define A n,l P n,l 1, 1. So, similar o he case for derangemens, 5.9 gives A n,l n n l ( l n, 1 l j! l! 2 F n j ) j l Now, define a n A n,,so n n! n a n per I J A ( n, j ) n j! j j 6.2 is he oal number of j-arrangemens of n for j, 1,...,n. Thus, we have a combinaorial inerpreaion of A n, Combinaorial Inerpreaion of A n,l We now give a combinaorial inerpreaion of A n,l for l 1. When l 1, recurrence 5.3 for P n,l 1, 1 gives A n,l A n,l 1 A n 1,l Proposiion 6.1. A n, a n, he oal number of arrangemens of n. In general, for l, A n,l is he number of arrangemens of n which conain {1, 2,...,l}. Proof. For l, le A denoe he se of arrangemens of n which conain l.wih, n,l we noe ha A n, is he se of all arrangemens. Le B n,l A n,l. We claim ha B n,l A n,l. The iniial values wih l are immediae. We show ha B n,l saisfies recurrence Consider A n,l 1.LeA A,soA is an arrangemen of n conaining l 1. If n,l 1 l 1, hen A A n, is any arrangemen. Now, eiher l A or l/ A. If l A, hen A A n,l, and so he number of arrangemens in A which conain l n,l 1 equals B n,l.

25 Inernaional Journal of Combinaorics 25 If l / A, hen by subracing 1 from all pars of A which are l 1, we obain an arrangemen of n 1 which conains l 1, ha is, an arrangemen in A n 1,l 1. Conversely, given an arrangemen in A n 1,l 1, adding 1 o all pars l yields an arrangemen in A n,l 1 which does no conain l. Hence, he number of arrangemens in A which do no conain n,l 1 l equals B n 1,l 1. We conclude ha B n,l 1 B n,l B n 1,l 1 ; hence, This is he resul. Example When n 3, we have a 3 A 3, 16 corresponding o he 16 arrangemens of 1 :, 1, 2, 3, 12, 21, 13, 31, 23, 32, 123, 132, 213, 231, 312, 321. Then, A 3,1 11 corresponding o he 11 arrangemens of 1 which conain {1}: 1, 12, 21, 13, 31, 123, 132, 213, 231, 312, 321. Then, A 3,2 8 corresponding o he 8 arrangemens of 1 which conain {1, 2}: 12, 21, 123, 132, 213, 231, 312, 321. Finally, A 3,3 3! 6 corresponding o he 6 arrangemens of 1 which conain {1, 2, 3}: 123, 132, 213, 231, 312, 321. Rearranging he facors in 5.9, we have n P n,l s, A ( j, l ) A ( n l, j l ) s n j j, j l 6.22 Wih s 1, his gives n A n,l A ( j, l ) A ( n l, j l ). j l 6.23 Here is a combinaorial explanaion of For any j l, oobainaj-arrangemen A of n conaining l, we may place he l poins of {1, 2,...,l} ino hese j posiions in A j, l ways. Then, he remaining j l posiions in A can be filled in by a j l -arrangemen of he unused n l poins in A n l, j l ways. Example Consider 5 A 5,2 A ( j, 2 ) A ( 3,j 2 ) j 2 A 2, 2 A 3, A 3, 2 A 3, 1 A 4, 2 A 3, 2 A 5, 2 A 3,

26 26 Inernaional Journal of Combinaorics Finally, from 2 of Theorem 5.3, s 1, and 1, we have A n,l l l 1 j a n j. j j 6.25 Example Consider ) 2 A 5,2 j 1 j( a 5 j a 5 a 4 j 1 2 a Permanens from I J Theorem 4.1 specializes o I J l min l,n l k A l,k JS nl k This can be wrien using he hypergeomeric form I J l min l,n l k k! 2 F ( k l, 1 k ) 1 JS nl k, 6.28 wih specrum ( α l, α n l 1 eigenvalue 2 F ), 1 n n occurring wih mulipliciy, α α by Corollary 4.3 and Proposiion 4.2.

27 Inernaional Journal of Combinaorics 27 Example For n 5, l 3, we have I J , wih characerisic polynomial λ 16 λ 11 4 λ As for he case of derangemens, he Johnson basis can be read off direcly from he marix I J l. Appendix Generalized Derangemen Numbers and Ineger Sequences The firs wo columns of he D n,l riangle, D n, and D n,1, give sequences A166 and A255 in he On-Line Encyclopedia of Ineger Sequences 7. The commens for A255 do no conain our combinaorial inerpreaion. The firs wo columns of he A n,l riangle, A n, and A n,1, give sequences A522 and A1339. The commens conain our combinaorial inerpreaion. The nex wo columns, A n,2 and A n,3, give sequences A134 and A134; here, our combinaorial inerpreaion is no menioned in he commens. Generalized Derangemen Triangles l is he lefmos column. The rows correspond o n from o 9.

28 28 Inernaional Journal of Combinaorics Values of D n,l A.1 Values of A n,l A.2 Exponenial polynomials h n,m s, Noe ha, as is common for marix indexing, we have dropped he commas in he numerical subscrips n h 1, h 1, h 2 2 2, h 3 6 3, h , A.3

29 Inernaional Journal of Combinaorics 29 n 1 h 1 s, h 11 s 2 2, h 12 2s 2 6 3, h 13 6s , h 14 24s , A.4 n 2 h 2 s 2 2s 2 2, h 21 s 2 4s 2 6 3, h 22 2s s , A.5 h 23 6s s , h 24 24s s , n 3 h 3 s 3 3s 2 6s 2 6 3, h 31 s 3 6s s , h 32 2s s s , A.6 h 33 6s s s , h 34 24s s s , n 4 h 4 s 4 4s 3 12s s , h 41 s 4 8s s s , h 42 2s s s s , h 43 6s s s s , A.7 h 44 24s s s s Acknowledgmen The auhors would like o hank Sacey Saples for discussions abou zeons and race formulas. References 1 P. J. Cameron, Permuaion Groups: London Mahemaical Sociey Suden Texs 45, Cambridge Universiy Press, Cambridge, UK, E. Bannai and I. Io, Algebraic Combinaorics I: Associaion Schemes, Benjamin/Cummings Publishing Company, San Francisco, Calif, USA, N. Biggs, Algebraic Graph Theory, Cambridge Universiy Press, Cambridge, UK, 2nd ediion, L. Come, Advanced Combinaorics: The Ar of Finie and Infinie Expressions, D. Reidel Publishing Company, Dordrech, Holland, J. Riordan, Inroducion o Combinaorial Analysis, Princeon Universiy Press, New Jersey, NJ, USA, B. R. Sonebridge, Derangemens of a mulise, Bullein of he Insiue of Mahemaics and Is Applicaions, vol. 28, pp , Online Encyclopedia of Ineger Sequences, hp://oeis.org/.