Combinatorial interpretations. Jacobi-Stirling numbers

Transcription

1 Combinatorial interpretations of Université Claude Bernard Lyon 1, France September 29th, th SLC

2 The Jacobi polynomials P (α,β) n (t) satisfy the Jacobi equation : (1 t 2 )y (t)+(β α (α+β+2)t)y (t)+n(n+α+β+1)y(t) = 0.

3 The Jacobi polynomials P (α,β) n (t) satisfy the Jacobi equation : (1 t 2 )y (t)+(β α (α+β+2)t)y (t)+n(n+α+β+1)y(t) = 0. Let l α,β [y](t) be the Jacobi differential operator : l α,β [y](t) = So, P n (α,β) (t) is a solution of 1 ( ) (1 t) α (1 + t) β (1 t) α+1 (1 + t) β+1 y (t) l α,β [y](t) = n(n + α + β + 1)y(t)

4 Everitt and al. gave the expansion of the n-th composite power of l α,β, involving a sequence of positive integers P (α,β) S(n, k) : n (1 t) α (1 + t) β l n α,β [y](t) = ( 1) k ( P (α,β) S(n, k)(1 t) α+k (1 + t) β+k y (k) ) (k) (t) k=0

5 Everitt and al. gave the expansion of the n-th composite power of l α,β, involving a sequence of positive integers P (α,β) S(n, k) : n (1 t) α (1 + t) β l n α,β [y](t) = ( 1) k ( P (α,β) S(n, k)(1 t) α+k (1 + t) β+k y (k) ) (k) (t) k=0 Actually, P (α,β) S(n, k) depends only on z = α + β + 1. We denote it by JS k n (z), the Jacobi-Stirling number of the second kind.

6 The JSn k (z) numbers are the relation coefficients in the following formula : n k 1 X n = JS k n(z) (X i(z + i)) k=0 i=0

7 The JSn k (z) numbers are the relation coefficients in the following formula : n k 1 X n = JS k n(z) (X i(z + i)) k=0 Equivalently, these numbers can be defined by the recurrence relation : i=0 JS k n(z) = JS k 1 n 1 (z) + k(k + z)jsk n 1(z), n, k 1 JS 0 0(z) = 1, JS k n(z) = 0 if k {1,..., n}

8 The Stirling numbers (of the second kind) S k n are defined by the relation : S k n = S k 1 n 1 + ks k n 1

9 The Stirling numbers (of the second kind) S k n are defined by the relation : S k n = S k 1 n 1 + ks k n 1 They count the number of : partitions of [n] := {1, 2,..., n} in k non-empty blocks. supdiagonal quasi-permutations of [n] := {1, 2,..., n} with k empty lines.

10 For example, is a partition of [6] in 3 blocks. π = { {1, 3, 6}, {2, 5}, {4} }

11 For example, π = { {1, 3, 6}, {2, 5}, {4} } is a partition of [6] in 3 blocks. It corresponds to the following quasi-permutation :

12 The central factorial numbers (of the second kind) U k n are defined by the relation : U k n = U k 1 n 1 + k2 U k n 1

13 The central factorial numbers (of the second kind) U k n are defined by the relation : They count the number of : U k n = U k 1 n 1 + k2 U k n 1 ordered pairs (π 1, π 2 ) of partitions of [n] in k blocks, with min(π 1 ) = min(π 2 ). ordered pairs (Q 1, Q 2 ) of supdiagonal quasi-permutations of [n], with Q 1 and Q 2 which have k same empty lines.

14 For example, if we note π 1 = { {1, 3, 6}, {2, 5}, {4} }, π 2 = { {1}, {2, 3, 5}, {4, 6} }, π 1 and π 2 are partitions of [6] in 3 blocks, with min(π 1 ) = min(π 2 )

15 For example, if we note π 1 = { {1, 3, 6}, {2, 5}, {4} }, π 2 = { {1}, {2, 3, 5}, {4, 6} }, π 1 and π 2 are partitions of [6] in 3 blocks, with min(π 1 ) = min(π 2 ) The ordered pair (π 1, π 2 ) corresponds to the folloqing ordered pair of supdiagonal quasi-permutations :

16 Let s come back to the : JS k n(z) = JS k 1 n 1 (z) + k(k + z)jsk n 1(z), n, k 1

17 Let s come back to the : JS k n(z) = JS k 1 n 1 (z) + k(k + z)jsk n 1(z), n, k 1 JS k n(z) is a polynomial in z of degree n k : JS k n(z) = a (0) n,k + a(1) n,k z + + a(n k) n,k z n k

18 Let s come back to the : JS k n(z) = JS k 1 n 1 (z) + k(k + z)jsk n 1(z), n, k 1 JS k n(z) is a polynomial in z of degree n k : JS k n(z) = a (0) n,k + a(1) n,k z + + a(n k) n,k z n k Moreover, a (n k) n,k = S k n a (0) n,k = Uk n

19 Definition A k-signed partition of [±n] 0 = {0, ±1, ±2,..., ±n} is a partition of [±n] 0 in k + 1 non-empty blocks B 0, B 1,..., B k, such that 0 B 0 et i [n], {i, i} B 0 j [k], i [n], {i, i} B j i = min B j [n] For example, π = { {0, 2, 5}, {±1, 2}, {±3}, {±4, 5} } is a 3-signed partition of [±5] 0.

20 Theorem a (i) n,k is equal to the number of k-signed partitions of [±n] 0 with i negative values in the block that contains 0.

21 Theorem a (i) n,k is equal to the number of k-signed partitions of [±n] 0 with i negative values in the block that contains 0. Proof : Since JS k n(z) verify the relation : JS k n(z) = JS k 1 n 1 (z) + k(k + z) JSk n 1(z), it suffices to check that the wanted numbers verify the recurrence : a (i) n,k = a(i) n 1,k 1 + ka(i 1) n 1,k + k2 a (i) n 1,k

22 Theorem a (i) n,k is equal to the number of k-signed partitions of [±n] 0 with i negative values in the block that contains 0. Proof : Since JS k n(z) verify the relation : JS k n(z) = JS k 1 n 1 (z) + k(k + z) JSk n 1(z), it suffices to check that the wanted numbers verify the recurrence : a (i) n,k = a(i) n 1,k 1 }{{} B k ={±n} + ka (i 1) n 1,k }{{} n B 0 + k 2 a (i) n 1,k }{{} other cases

23 a (i) n,k = {k signed partitions of [±n] 0 with i values < 0 in B 0 }

24 a (i) n,k = {k signed partitions of [±n] 0 with i values < 0 in B 0 } For i = n k, we recover the interpretation of S k n. π = { {0, 3, 5, 6}, {±1, 3, 6}, {±2, 5}, {±4} } π = { {1, 3, 6}, {2, 5}, {4} }

25 a (i) n,k = {k signed partitions of [±n] 0 with i values < 0 in B 0 } For i = n k, we recover the interpretation of S k n. π = { {0, 3, 5, 6}, {±1, 3, 6}, {±2, 5}, {±4} } π = { {1, 3, 6}, {2, 5}, {4} } For i = 0, we recover the interpretation of U k n. π = { {0, 3, 6}, {±1, 3, }, {±2, 5}, {±4, 5, 6} } π 1 = { {1, 3}, {2}, {4, 5, 6} }, π 2 = { {1, 3}, {2, 5}, {4, 6} }

26 Definition A simply hooked k-quasi-permutation of [n] (k-shqp of [n]) is a part Q of a tableau [n] [n] such that : Q is contained in the graph of a permutation σ without fix points, each diagonal hook contains at most one elemnt, there are k empty diagonal hooks. For example, is a 2-SHQP of [8].

27 Definition A simply hooked k-quasi-permutation of [n] (k-shqp of [n]) is a part Q of a tableau [n] [n] such that : Q is contained in the graph of a permutation σ without fix points, each diagonal hook contains at most one elemnt, there are k empty diagonal hooks. For example, is a 2-SHQP of [8].

28 Definition A simply hooked k-quasi-permutation of [n] (k-shqp of [n]) is a part Q of a tableau [n] [n] such that : Q is contained in the graph of a permutation σ without fix points, each diagonal hook contains at most one elemnt, there are k empty diagonal hooks. For example, is a 2-SHQP of [8].

29 Definition A simply hooked k-quasi-permutation of [n] (k-shqp of [n]) is a part Q of a tableau [n] [n] such that : Q is contained in the graph of a permutation σ without fix points, each diagonal hook contains at most one elemnt, there are k empty diagonal hooks. For example, is a 2-SHQP of [8].

30 Definition A simply hooked k-quasi-permutation of [n] (k-shqp of [n]) is a part Q of a tableau [n] [n] such that : Q is contained in the graph of a permutation σ without fix points, each diagonal hook contains at most one elemnt, there are k empty diagonal hooks. For example, is a 2-SHQP of [8].

31 Definition A simply hooked k-quasi-permutation of [n] (k-shqp of [n]) is a part Q of a tableau [n] [n] such that : Q is contained in the graph of a permutation σ without fix points, each diagonal hook contains at most one elemnt, there are k empty diagonal hooks. For example, is a 2-SHQP of [8].

32 Definition A simply hooked k-quasi-permutation of [n] (k-shqp of [n]) is a part Q of a tableau [n] [n] such that : Q is contained in the graph of a permutation σ without fix points, each diagonal hook contains at most one elemnt, there are k empty diagonal hooks. For example, is a 2-SHQP of [8].

33 Definition A simply hooked k-quasi-permutation of [n] (k-shqp of [n]) is a part Q of a tableau [n] [n] such that : Q is contained in the graph of a permutation σ without fix points, each diagonal hook contains at most one elemnt, there are k empty diagonal hooks. For example, is a 2-SHQP of [8].

34 Definition A simply hooked k-quasi-permutation of [n] (k-shqp of [n]) is a part Q of a tableau [n] [n] such that : Q is contained in the graph of a permutation σ without fix points, each diagonal hook contains at most one elemnt, there are k empty diagonal hooks. For example, is a 2-SHQP of [8].

35 Theorem a (i) n,k counts also the number of ordered pairs (Q 1, Q 2 ) of k-shqp of [n] such that Q 1 and Q 2 have k empty lines (the same), Q 1 and Q 2 have identical subdiagonal parts, Q 1 and Q 2 have i filled boxes in their subdiagonal parts.

36 Pour i = n k, we recover the interpretation of S k n.

37 Pour i = n k, we recover the interpretation of S k n. For i = 0, we recover the interpretation of U k n.

38 The (of the first kind) js k n(z) are defined by inversing the relation on the JS k n(z) numbers : X n = n k=0 n 1 (X i(z + i)) = i=0 k 1 JS k n(z) (X i(z + i)) i=0 n ( 1) n k js k n(z)x k k=0

39 The (of the first kind) js k n(z) are defined by inversing the relation on the JS k n(z) numbers : X n = n k=0 n 1 (X i(z + i)) = i=0 k 1 JS k n(z) (X i(z + i)) i=0 n ( 1) n k js k n(z)x k k=0 It follows that they verify the following relation : js k n(z) = js k 1 n 1 (z) + (n 1)(n 1 + z) jsk n 1(z)

40 The Stirling numbers (of the first kind) s k n are defined by the relation : s k n = s k 1 n 1 + (n 1)sk n 1 The central factorial numbers (of the first kind) u k n are defined by the relation : u k n = u k 1 n 1 + (n 1)2 u k n 1

41 The Stirling numbers (of the first kind) s k n are defined by the relation : s k n = s k 1 n 1 + (n 1)sk n 1 The central factorial numbers (of the first kind) u k n are defined by the relation : u k n = u k 1 n 1 + (n 1)2 u k n 1 s k n is the number of permutations of [n] with k cycles. u k n had still no interpretation until present.

42 js k n(z) is a polynomial in z of degree n k : js k n(z) = b (0) n,k + b(1) n,k z + + b(n k) n,k z n k

43 js k n(z) is a polynomial in z of degree n k : js k n(z) = b (0) n,k + b(1) n,k z + + b(n k) n,k z n k Moreover, b (n k) n,k = s k n b (0) n,k = uk n

44 Definition Let w = w(1)w(2)... w(l) be a word on the alphabet [n]. A letter w(j) is a record of w if w(j) < w(k), k = 1... j 1 We denote by rec(w) the number of records of w and rec 0 (w) = rec(w) 1. For example, for the records are w = , 5, 4, 2, 1. Thus, rec(w) = 4 and rec 0 (w) = 3.

45 Theorem b (i) n,k is the number of ordered pairs (σ, τ) with : σ is a permutation of [n] {0} with k cycles, τ is a permutation of [n] with k cycles, σ and τ have the same cyclic minimas, 1 Orb σ (0) and rec 0 (w) = i where w = σ(0)... σ l (0) with σ l+1 (0) = 0.

46 Theorem b (i) n,k is the number of ordered pairs (σ, τ) with : σ is a permutation of [n] {0} with k cycles, τ is a permutation of [n] with k cycles, σ and τ have the same cyclic minimas, 1 Orb σ (0) and rec 0 (w) = i where w = σ(0)... σ l (0) with σ l+1 (0) = 0. Idea : interpret the formula b (i) n,k = b(i) n 1,k 1 + (n 1)b(i 1) n 1,k + (n 1)2 b (i) n 1,k

47 For i = n k, we recover the interpretation of s k n.

48 For i = n k, we recover the interpretation of s k n. For i = 0, we recover the interpretation of u k n. u k n is the number of ordered pairs of permutations (σ 1, σ 2 ) of [n] with k cycles, with same cyclic minimas.

49 Thanks for your attention.