Some multivariable master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions a

Transcription

1 ESI, Universität Wien Some multivariable master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions a Jiang Zeng Université Lyon, France October 7, 207, Wien a Joint work with Alan Sokal (New York/London)

2 S-type and J-type continued fractions If (a n ) n 0 is a sequence of combinatorial numbers or polynomials with a 0 =, it is often fruitful (total positivity of (a i+j ) i,j 0, log-convexity, γ-positivity, moment sequence,...) to seek to express its ordinary generating function (OGF) as a continued fraction of either Stieltjes (S) type, a n t n =, α t n=0 or Jacobi type (J), a n t n = n=0 γ 0 t α 2t β t 2 γ t β 2t 2,

3 Contraction formulae of an S-fraction to a J-fraction i.e., α x α 2x = α α 2 x 2. α x (α 2 + α 3 )x α 3α 4 x 2 γ 0 = α γ n = α 2n + α 2n+ for n β n = α 2n α 2n.

4 Two approaches This line of investigation, i.e. (a n ) (α n ) (or ((γ n ), (β n ))), goes back at least to Euler, but it gained impetus following Flajolet s seminal discovery that any S-type (resp. J-type) continued fraction can be interpreted combinatorially as a generating function of Dyck (resp. Motzkin) paths with suitable weights for each rise and fall (resp. each rise, fall and level step).

5 Two approaches This line of investigation, i.e. (a n ) (α n ) (or ((γ n ), (β n ))), goes back at least to Euler, but it gained impetus following Flajolet s seminal discovery that any S-type (resp. J-type) continued fraction can be interpreted combinatorially as a generating function of Dyck (resp. Motzkin) paths with suitable weights for each rise and fall (resp. each rise, fall and level step). Our approach will be (in part) to run this program in reverse: we start from a continued fraction in which the coefficients α (or γ and β) contain indeterminates in a nice pattern, and we attempt to find a combinatorial interpretation for the resulting polynomials a n.

6 Euler s continued formulae n!x n = n 0 = x with coefficients α 2k = k, α 2k = k. x x 2x 2x 2 x 2 3x 22 x 2

7 A naive generalization Introduce the polynomials P n (x, y, u, v) by the following CF P n (x, y, u, v)t n = n 0 x t. () y t (x + u) t (y + v) t with coefficients α 2k = x + (k )u α 2k = y + (k )v. (2) Clearly P n (x, y, u, v) is a homogeneous polynomial of degree n and P n (,,, ) = n!.

8 Record classification Given a permutation S n, an index i [n] (or value σ(i) [n]) is called a record (rec) (or left-to-right maximum) if σ(j) < σ(i) for all j < i; antirecord (arec) (or right-to-left minimum) if σ(j) > σ(i) for all j > i; exclusive record (erec) if it is a record and not also an antirecord; exclusive antirecord (earec) if it is an antirecord and not also a record; record-antirecord (rar) if it is both a record and an antirecord; neither-record-antirecord (nrar) if it is neither a record nor an antirecord.

9 Cycle classification We say that an index i [n] is a cycle peak (cpeak) if σ (i) < i > σ(i); cycle valley (cval) if σ (i) > i < σ(i); cycle double rise (cdrise) if σ (i) < i < σ(i); cycle double fall (cdfall) if σ (i) > i > σ(i); fixed point (fix) if σ (i) = i = σ(i). We denote the number of cycles, records, antirecords,... in σ by cyc(σ), rec(σ), arec(σ),..., respectively. A rougher classification is that an index i [n] (or value σ(i)) is an excedance (exc) if σ(i) > i; anti-excedance (aexc) if σ < i; fixed point (fix) if σ = i.

10 Two combinatorial interpretations Theorem The polynomials defined by the S-fraction have the combinatorial interpretations P n (x, y, u, v) = σ S n x arec(σ) y erec(σ) u n exc(σ) arec(σ) v exc(σ) erec(σ) (3) and P n (x, y, u, v) = σ S n x cyc(σ) y erec(σ) u n exc(σ) cyc(σ) v exc(σ) erec(σ). (4) N.B. The triple statistics (arec, erec, exc) and (cyc, erec, exc) are equidistributed on S n.

11 Special cases () The Stirling cycle polynomials n P n (x,,, ) = S(n, k)x k = x(x + )... (x + n ). k=0 or their homogenized version n P n (x, y, y, y) = S(n, k)x k y n k = x(x+y)... (x+(n )y). k=0 The Eulerian polynomials n P n (, y,, y) = A n (y) = A(n, k)y k or P n (x, y, x, y) = A n (x, y) = k=0 n A(n, k)x n k y k. k=0

12 Special cases (2): Dumont-Kreweras 988 The record-antirecord permutation polynomials P n (a, b,, ) = σ S n a arec(σ) b erec(σ) or Note that P n (a, b, c, c) = σ S n a arec(σ) b erec(σ) c n arec(σ) erec(σ). P n (a, b,, )t n = n=0 2 F 0 (a, b; t) 2F 0 (a, b ; t).

13 Special cases (3) The polynomials [sequence A45879/A202992] P n (x, x, u, u) = n T (n, k)x n k u k k=0 where T (n, k) is the number of permutations in S n having exactly k indices that are the middle point of a pattern 32 (or 23). In particular T (n, 0) is the number of 23-avoiding permutations, which equals the Catalan number C n. So the polynomials interpolate between C n and n!.

14 Special cases (4): Narayanan polynomials P n (x, y, 0, 0) = σ S n(23) = = σ S n(23) n k=0 n ( n k x arec(σ) y erec(σ) x cyc(σ) y erec(σ) )( n k ) x k y n k. The cycle interpretation (with y = ) was given by Parviainen (2007).

15 Record and cycle classifications We have classified indices in a permutation according to their record status: exlusive record, exclusive antirecord, record-antirecord or neither-record-antirecord and aslo according to their cycle status: cycle peak, cycle valley, cycle double rise, cycle double fall or fixed point. Applying now both classifications simultaneously, we obtain 0 disjoint categories.

16 Record-cycle classifications: 0 classes ereccval: exclusive records that are also cycle valleys; erecdrise: exclusive records that are also cycle double rises; eareccpeak: exclusive antirecords that are also cycle peaks; eareccdfall: exclusive antirecords that are also cycle double falls; rar: record-antirecords (that are always fixed points); nrcpeak: neither-record-antirecords that are also cycle peakss; nrcval: neither-record-antirecords that are also cycle valleys; nrcdrise: neither-record-antirecords that are also cycle double falls; nrcfall: neither-record-antirecords that are also cycle falls; nrfix: neither-record-antirecords that are also fixed points.

17 First J-fraction Q n (x, x 2,y, y 2, z, u, u 2, v, v 2, w) = x eareccpeak(σ) x earccdfall(σ) 2 y ereccval(σ) y ereccdrise(σ) 2 z rar(σ) σ S n u nrcpeak(σ) u nrcdfall(σ) 2 v nrcval(σ) v nrcdrise(σ) 2 w nrfix(σ) If i is a fixed point of σ, we define its level by lev(i, σ) := #{j < i : σ(j) > i} = #{j > i : σ(j) < i}. Clearly, a fixed point i is a record-antirecord if its level is 0, and a neither-record-antirecord if its level is.

18 First master polynomial Introduce indeterminates w = (w l ) l 0 and write w fix(σ) := l=0 w fix(σ,l) l = i Fix(σ) w lev(i,σ). The master polynomial encoding all these (now infinitely many) statistics is Q n (x, x 2,y, y 2, u, u 2, v, v 2, w) = x eareccpeak(σ) x earccdfall(σ) 2 y ereccval(σ) y ereccdrise(σ) 2 σ S n u nrcpeak(σ) u nrcdfall(σ) 2 v nrcval(σ) v nrcdrise(σ) 2 w fix(σ)

19 Theorem 2 (First J-fraction for permutations) The OGF of the polynomials Q n has the J-type continued fraction Q n (x, x 2, y, y 2, u, u 2, v, v 2, w)t n = n=0 x y t 2, w 0 t (x 2 + y 2 + w )t (x + u )(y + v )t 2 with coefficients γ 0 = w 0, γ n = [x 2 + (n )u 2 ] + [y 2 + (n )v 2 ] + w n for n β n = [x + (n )u ][y + (n )v ].

20 Second J-fraction Define the polynomial ˆQ n (x, x 2, y, y 2, u, u 2, v, v 2, w, λ) = x eareccpeak(σ) x earccdfall(σ) 2 y ereccval(σ) y ereccdrise(σ) 2 σ S n u nrcpeak(σ) u nrcdfall(σ) 2 v nrcval(σ) v nrcdrise(σ) 2 w fix(σ) λ cyc(σ).

21 Second J-fraction Theorem 3 (Second J-fraction for permutations) The OGF of the polynomials Q n has the J-type continued fraction ˆQ n (x, x 2, y, y 2, u, u 2, y, y 2, w, λ)t n = n=0 λx y t 2, λw 0 t (x 2 + y 2 + λw )t (λ + )(x + u )t 2 with coefficients γ 0 = λw 0, γ n = [x 2 + (n )u 2 ] + ny 2 + λw n for n β n = (λ + n )[x + (n )u ]y.

22 Statistics on permutations () Comparing Theorem () with the first J-fraction the polynomial Q n reduces to P n (x, y, u, v) if we set x = x 2 = x, y = y 2 = y, w 0 = xz u = u 2 = w l = (l ), v = v 2 = v. The weight function reduces to w(σ) = x arec(σ) y erec(σ) v exc(σ) z rar(σ). Comparing with Theorem (2) with the second J-fraction the polynomial ˆQn reduces to P n (x, y, u, v) if we set x = x 2 = y, u = u 2 = v, w 0 = z y = y 2 = v = v 2 = w l = (l ), λ = x. The weight function reduces to ŵ(σ) = x cyc(σ) y earec(σ) v aexc(σ) z rar(σ).

23 Statistics on permutations (2) We have the following equidistribution: (arec, erec, exc, rar) (cyc, erec, exc, rar). Cori (2008) and Foata-Han (2009) : (cyc, arec) (rec, arec) on S n and the distribution of (cyc, arec) is symmetric. Kim-Stanton (205): (rec, arec, rar) moments of associated Laguerre polynomials. Elizalde (207): (cyc, fix, aexc, cdfall), which is (cyc, fix, exc, cdrise) by σ σ.

24 Setting v = and z = y we have ( ) x cyc(σ) y arec(σ) t n = n=0 σ S n xy t xy t (x + )(y + ) t (x + y + ) t with γ 0 = xy, γ n = x + y + 2n β n = (x + n )(y + n ) for n.

25 p,q-generalizations of Euler s continued fractions We define [n] p,q = pn q n p q p j q n j. = n Foata-Zeilberger (990), Biane (993), De Mdicis-Viennot (994), Simion-Stanton(994, 996), Clarke-Steingrimsson-Z. (997), Randrianarivony (998), Corteel (2007),... Let [n; a] p,q = ap n + p n 2 q + + pq n 2 + q n. Then Randrianarivony (998) : j=0 γ n = (a[n + ; α] r,s + b[n; β] t,u )x n, β n = cd[n; γ] p,q [n; µ] v,w x 2n.

26 Crossings and nestings To the permutation π = (, 9, 0, 2, 3, 7)(4)(5, 6, )(8) S we associate a pictorial representation as follows:

27 Upper and lower crossings and nestings We say that a quadruple i < j < k < l forms an upper crossing (ucross) if k = σ(i) and l = σ(j); lower crossing (lcross) if i = σ(k) and j = σ(l); upper nesting (unest) if l = σ(i) and k = σ(j); lower nesting (lnest) if i = σ(l) and j = σ(k). We consider also some degenerate cases with j = k, by saying a triplet i < j < k forms an upper joining (ujoin) if j = σ(i) and l = σ(j); lower joining (lcross) if i = σ(j) and j = σ(l); upper pseudo-nesting (upsnest) if l = σ(i) and j = σ(j); lower pseudo-nesting (lpsnest) if i = σ(l) and j = σ(j).

28 Refinement of crossing categories We say that a quadruplet i < j < k < l forms an upper crossing of type cval (ucrosscval) if k = σ(i) and l = σ(j) and σ (j) > j; upper crossing of type cdrise (ucrosscdrise) if k = σ(i) and l = σ(j) and σ (j) < j; lower crossing of type cpeak lcrosscpeak) if l = σ(i) and σ (k) < k; lower crossing of type cdfall (lcrosscdfall) if i = σ(k) and j = σ(l) and σ (k) > k;

29 Refinement of nesting categories We say that a quadruplet i < j < k < l forms an upper nesting of type cval (unestcval) if l = σ(i) and k = σ(j) and σ (j) > j; upper nesting of type cdrise (unestcdrise) if l = σ(i) and k = σ(j) and σ (j) < j; lower nesting of type cpeak (lnestcdpeak) if l = σ(i) and j = σ(j) and σ (k) < k; lower nesting of type cdfall (lnestcdfall) if i = σ(l) and j = σ(j) and σ (k) > k.

30 First J-fraction for permutations Define the polynomial Q n (x, y, u, v, w, p, q, s) := Q n (x, x 2, y, y 2, u, u 2, v, v 2, w, p +, p +2, p +2, p, p 2, q +, q +2, q, q 2, s) = x eareccpeak(σ) x earccdfall(σ) 2 y ereccval(σ) y ereccdrise(σ) 2 σ S n u nrcpeak(σ) u nrcdfall(σ) 2 v nrcval(σ) v nrcdrise(σ) 2 w fix(σ) p ucrosscval(σ) + p ucrosscdrise(σ) q unestcval(σ) + q unestcdrise(σ) +2 p lcrosscpeak(σ) +2 q lnestcpeak(σ) p lcrosscdfall(σ) 2 q inestcdfall(σ) 2 s psnest(σ).

31 First J-fraction for permutations 2 Q n (x, y, u, v, w, p, q, s)t n = n=0 w 0 t x y t 2 (x 2 + y 2 + sw )t (p x + q u )(p + y + q + v )t 2 with coefficents γ 0 = w 0 and for n, γ n = (p n 2 x 2 + q 2 [n ] p 2,q 2 u 2 ) + (p n +2 y 2 + q +2 [n ] p+2,q +2 v 2 ) + s n w n β n = (p n x + q [n ] p,q u )(p n + y + q + [n ] p+,q + v ).

32 First master J-fraction () Rather than counting the total numbers of nestings, we should instead count the number of upper (resp. lower) crossings or nestings that use a particular vertex j (resp. k) in second (resp. third) position, and then attribute weights to the vertex j (resp. k) depending on these values. ucross(j, σ) = #{i < j < k < l : k = σ(i) and l = σ(j)} unest(j, σ) = #{i < j < k < l : k = σ(j) and l = σ(i)} lcross(k, σ) = #{i < j < k < l : i = σ(k) and j = σ(l)} lnest(k, σ) = #{i < j < k < l : i = σ(l) and j = σ(k)}.

33 First master J-fraction (2) N.B. ucross(j, σ) and unest(j, σ) can be nonzero only when j is a cycle valley or a cycle double rise, while lcross(k, σ) and lnest(k, σ) can be nonzero only when k is a cycle peak or a cycle double fall. And obviously we have ucrosscval(σ) = ucross(j, σ) j cval and analogously for the other seven crossing/nesting quantities.

34 First master J-fraction (3) We now introduce five infinite families of indeterminates a, b, c, d where x = (x l,l ) l,l 0 and w = (w l ) l 0, and define the polynomial Q n (a, b, c, d, w) = a ucross(i,σ),unest(i,σ) σ S n i cval i cpeak c lcross(i,σ),lnest(i,σ) i cdfall i cdrise b lcross(i,σ),lnest(i,σ) d ucross(i,σ),unest(i,σ) These polynomials then have a beautiful J-fraction. i fix w lev(i,σ)

35 First master J-fraction (4) Theorem 4 (First master J-fraction for permutations) The OGF of the polynomials Q n (a, b, c, d, w) has the J-type continued fraction Q n (a, b, c, d, w)t n = n=0 w 0 t a 00 b 00 t 2 (c 00 + d 00 + w )t (a 00 + a 0 )(b 0 + b 0 )t 2 with coefficients γ n = c n + d n + w n and β n = a n b n, where a n := n l=0 a l,n l = a 0,n + a,n a n,0.

36 A remark on the inversion statistic A inversion of a permutation σ S n is a pair i, j [n] such that i < j and σ(i) > σ(j). Lemma (Shin-Z. 200) We have inv = cval + cdrise + cdfall + ucross + lcross + 2(unest + lnest + psnest) = exc +(ucross + lcross + ljoin) + 2(unest + lnest + psnest).

37 p,q-generalization of the second J-fraction We can also make a (p,q)-generalization of the second J-fraction involving cyc. Define the polynomial ˆQ n (x, x 2, y, y 2, u, u 2, v, v 2, w, p +, p +2, p, p 2, q +, q +2, q, q 2, s, λ) = x eareccpeak(σ) x earccdfall(σ) 2 y ereccval(σ) y ereccdrise(σ) 2 σ S n u nrcpeak(σ) u nrcdfall(σ) 2 v nrcval(σ) v nrcdrise(σ) 2 w fix(σ) p ucrosscval(σ) + p ucrosscdrise(σ) q unestcval(σ) + q unestcdrise(σ) +2 p lcrosscpeak(σ) +2 q lnestcpeak(σ) p lcrosscdfall(σ) 2 q lnestcdfall(σ) 2 s psnest(σ) λ cyc(σ).

38 p,q-generalization of the second J-fraction Theorem 5 (Second J-fraction for permutations) n=0 ˆQ n (x, x 2, y, y 2, u, u 2, v, v 2, w, p +, p +2, p, p 2, q +, q +2, q, q 2, s, λ)t n = λw 0 t with coefficients γ 0 = λw 0, λx y t 2 (x 2 + y 2 + λw )t (λ + )(x + u )t 2 γ n = (p 2 n x 2 + q 2 [n ] p 2,q 2 u 2 ) + np+2 n y 2 + λs n w n for n β n = (p n x + q [n ] p,q u )p+ n y (λ + n ).,

39 Set partitions: S-fraction The Bell number B n is the number of partitions of an n-element set into nonempty blocks with B 0 =. B n t n = n=0 t t t 2 t with coefficients α 2k =, α 2k = k.

40 B n (x, y, v) t n = n=0 x t y t x t (y + 2v) t with coefficients α 2k = x, α 2k = y + (k )v. Clearly B n (x, y, v) is a homogeneous polynomial of degree n; it has three truly independent variables.

41 Theorem 6 (S-fraction for set) The polynomials B n (x, y, v) have the combinatorial interpretation B n (x, y, v) = π Π n x π y erec(π) v n π erec(π) where π (resp. erec(π)) denotes the number of blocks (resp. exclusive records) in π. Given π Π n, for i [n], we define σ (i) to be the next-larger element after i in its block, if i is not the largest element in its block, and 0 otherwise. Then erec(π) := erec(σ ). For example, if π = {, 5} {2, 3, 7} {4} {6}, then σ =

42 Given a partition π of [n], we say that an element i [n] is an opener if it is the smallest element of a block of size 2; a colser if it is the largest element of a block of size 2; an insider if it is a non-opener non-closer element of a block of size 3; a singleton if it is the sole element of a block of size. Clearly every element i [n] belongs to precisely one of these four classes.

43 J-fraction We can refine the polynomial B n (x, y, v) by distinguishing between singletons and blocks of size 2; in addition, we can distinguish between exclusive records that are openers and those that are insiders. Define B n (x, x 2, y, y 2, v) = x m(π) x m 2(π) 2 π Π n y erecin(π) y erecop(π) 2 v n π erec(π), where m (π) is the number of singletons in π, m 2 (π) is the number of non-singletons blocks, erecin(π) is the number of exclusive records that are insiders, and erecop(π) is the number of exclusive records that are openers.

44 Theorem 7 (J-fraction for set partitions) B n (x, x 2, y, y 2, v)t n = n=0 x 2 y 2 t 2 x t (x + y )t x 2(y 2 + v)t 2 with coefficients γ 0 = x, γ n = x + y + (n )v for n β n = x 2 [y 2 + (n )v].

45 First p, q generalization Let π = {B, B 2,..., B k } be a partition of [n]. We associate a graph G π with vertex set [n] such that i, j are joined by an edge if and only if they are consecutive elements within the same block. We then say that a quadruplet i < j < k < l forms a crossing (cr) if (i, k) G π and (j, l) G π ; nesting (ne) if (i, l) G π and (j, k) G π. We also say that a triplet i < k < l forms a pseudo-nesting (psne) if (i, l) G π π = {{, 9, 0}, {2, 3, 7}, {4}, {5, 6, }, {8}}.

46 We now introduce a (p, q)-generalization of previous polynomial: B n (x, x 2, y, y 2, v, p, q, r) = x m(π) x m 2(π) 2 y erecin(π) y erecop(π) 2 π Π n v n π erecop(π) p cr(π) q ne(π) r psne(π). Theorem 8 B n (x, x 2, y, y 2, v, p, q, r)t n = n=0 with coefficients γ 0 = x, x t x 2y 2 t 2 γ = r n x + p n y + q[n ] p,q v for n β n = x 2 (p n y 2 + q[n ] p,q v).

47 First master J-fraction Rather that counting the total numbers of quadrauplets i < j < k < l that form crossings or nestings, we should instead count the number of crossings or nestings that use a particular vertex k in third (or sometimes second) position, and then attribute weights to the vertex k depending on those values. We define cr(k, π) = #{i < j < k < l : (i, k) G π and (j, l) G π } ne(k, π) = #{i < j < k < l : (i, l) G π and (j, k) G π } psne(k, π) = #{i < k < l : k is a singleton and (i, l) G π }

48 First master J-fraction Note that cr(π) and ne(k, π) can be nonzero only when k is either an insider or a closer; and we obviously have cr(π) = cr(k, π) Finally we define ne(π) = psne(π) = k insiders closers k insiders closers k singletons ne(k, π) psne(k, π). crne (π) = #{i < k < l : k is an opener and (i, l) G π } which counts the number of times that the opener k occurs in second position in a crossing or nesting.

49 First master J-fraction We now introduce four infinite families of indterminates a = (a l ) l 0, b = (a l,l ) l,l 0, c = (c l,l ) l,l 0, e = (e l ) l 0 and define the polynomials B n (a, b, c, e) by B n (a, b, c, e) = a crne (i,π) b cr(i,π),ne(i,π) π Π n i openers i closers c cr(i,π),ne(i,π) e psne(i,π) i insiders i singletons

50 Theorem 9 (Master J-fraction for set partitions) The OGF of the polynomials B n (a, b, c, e) has the J-type CF B n (a, b, c, e)t n = n=0 with coefficients e 0 t a 0 b 00 t 2 (c 00 + e )t a (b 0 + b 0 )t 2 γ n = n l=0 c l,n l + e n, β n = a n n l=0 b l,n l.

51 Perfect matchings Euler: (2n )!!t n = n=0 t 2 t 3 t We introduce the polytnomials M n (x, y, u, v) by M n (x, y, u, v)t n = x t n=0 (x + v) t (x + 2u) t with coefficients α 2k = x + (2k 2)u, α 2k = y + (2k )v

52 Master S-fraction We can regard a perfect matching either as a special type of partition (namely, one in which all blocks are of size 2) or as a special type of permutation (namely, a fixed-point-free involution). We now introduce four infinite families of indterminates a = (a l ) l 0, b = (a l,l ) l,l 0,and define the polynomials M n (a, b) by M n (a, b) = b cr(i,π),ne(i,π). a crne (i,π) π M n i openers i closers Of course, we have M n (a, b) = B 2n (a, b, 0, 0).

53 Theorem 0 (Master S-fraction for perfect matchings) The OGF of the polynomials B n (a, b) has the S-type CF M n (a, b)t n = n=0 a 0 b 00 t 2 a (b 0 + b 0 )t 2 with coefficients α n = a n b n, where b n = n l=0 b l,n l.