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### Division and the Giambelli Identity [article]

Susan Y. J. Wu, Arthur L. B. Yang
2005 arXiv   pre-print
{B 1 , B 2 , . . . , B n }, and I, J ∈ N n , then the multi-Schur function of index J/I is defined as follows [6] : S J/I (A 1 − B 1 ; . . . ; A n − B n ) := S j k −i l +k−l (A k − B k ) 1≤l,k≤n . (3  ...  Given two sets of variables A and B, denote by R(A, B) the product a∈A,bB (a − b), and by A − B the set difference.  ...

### Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences [article]

William Y. C. Chen, Arthur L. B. Yang, Elaine L. F. Zhou
2010 arXiv   pre-print
≤ 1. (2.2) In order to show that (x + 1)B(x) is ratio monotone, we need to justify that b 2n+1 b 0 ≤ b 2n b 1 ≤ · · · ≤ b 2n+1−i b i ≤ · · · ≤ b n+1 b n ≤ 1 (2.3) and b 0 b 2n ≤ b 1 b 2n−1 ≤ · · · ≤ b  ...  So we have b m b 0 − a 0 ≤ b m−1 b 1 ≤ · · · ≤ b m−i b i ≤ · · · ≤ b m−[ m−1 2 ] b [ m−1 2 ] ≤ 1 (2.8) and b 0 − a 0 b m−1 ≤ b 1 b m−2 ≤ · · · ≤ b i−1 b m−i ≤ · · · ≤ b [ m ≤ 1. (2.9) Clearly, (2.6) follows  ...

### The srank Conjecture on Schur's Q-Functions [article]

William Y. C. Chen, Donna Q. J. Dou, Robert L. Tang, Arthur L. B. Yang
2008 arXiv   pre-print
Let Q (a) = q a and Q (a,b) = q a q b + 2 b m=1 (−1) m q a+m q b−m . From (3.7) we see that Q (a,b) = −Q (b,a) and thus Q (a,a) = 0 for any a, b.  ...  A BB t 0 , where A = (Q (λ i ,λ j ) ) and B = (Q (λ i −µ n+1−j ) ) .  ...  Let us consider the rows without 0's, and there are two possibilities: (A) o * r ≥ e * r , (B) o * r < e * r .  ...

### The Zrank Conjecture and Restricted Cauchy Matrices [article]

Guo-Guang Yan, Arthur L. B. Yang, Joan J. Zhou
2005 arXiv   pre-print
The length (S) of a snake S is defined to be one less than its number of squares. For an empty snake S, let (S) = −1.  ...  From Figure 1 , we see that SS((7, 6, 6, 3)/(3, 1)) = L 0 L 1 OOOOL 2 R 2 R 1 OR 0 . Let rank(λ/µ) = r, and let SS(λ/µ) = q 1 q 2 · · · q k .  ...  Let s ∈ {b 1 , . . . , b r−1 } and s ′ ∈ {a 3 , . . . , a r , b r }. Then we have s < s ′ and It follows that f (a 1 ) < f (a 2 ), namely N < 0.  ...

### Branden's Conjectures on the Boros-Moll Polynomials [article]

William Y. C. Chen, Donna Q. J. Dou, Arthur L. B. Yang
2012 arXiv   pre-print
Define L to be an operator acting on the sequence {a i } n i=0 as given by L({a i } n i=0 ) = {b i } n i=0 , where b i = a 2 i − a i+1 a i−1 for 0 ≤ i ≤ n under the convention that a −1 = 0 and a n+1 =  ...  Furthermore, Chen, Yang and Zhou [8] proved that if f (x) is a polynomial with nondecreasing and nonnegative coefficients, then f (x + 1) is ratio monotone.  ...

### On the Modes of Polynomials Derived from Nondecreasing Sequences [article]

Donna Q. J. Dou, Arthur L. B. Yang
2010 arXiv   pre-print
Chen, Yang and Zhou [4] showed that P (x + 1) is ratio monotone, which leads to an alternative proof of the ratio monotonicity of the Boros-Moll polynomials [3] .  ...  Therefore, f (j+1) k (x) = (k + j − 1) j b ′ k+j−1 (x) − (k + j) j b ′ k+j (x) = (k + j)(k + j − 1) j b k+j (x) − (k + j + 1)(k + j) j b k+j+1 (x) = (k + j) j+1 b k+j (x) − (k + j + 1) j+1 b k+j+1 (x).  ...

### A Blass-Sagan bijection on Eulerian equivalence classes [article]

Beifang Chen, Arthur L. B. Yang, Terence Y. J. Zhang
2007 arXiv   pre-print
By definitions, the two orientations (B-1) and (B-2) in Fig. 1 are cut equivalent, (B-2) and (B-3) are Eulerian equivalent, while (B-1) and (B-3) are Eulerian-cut equivalent.  ...  (b) ε is totally cyclic. (c) ε is reduced.  ...

### Stable Equivalences of Giambelli Type Matrices of Schur Functions [article]

William Y. C. Chen, Arthur L. B. Yang
2005 arXiv   pre-print
Figure 2 2 Figure 2 . 1 21 The cutting strip of an outside decomposition Figure 2 2 Figure 2 . 2 22 Four possible types of diagonals of λ/µ Figure 2 . 2 3 ω i acts on a Type 1 diagonal L (b) If  ...  Chen, Yan and Yang [1] obtained the following characterization of outside decompositions of a skew shape.  ...

### Transformations of Border Strips and Schur Function Determinants [article]

William Y. C. Chen, Guo-Guang Yan, Arthur L. B. Yang
2004 arXiv   pre-print
Suppose b i 1 < b i 2 < . . . < b im is the reordering of b 1 , b 2 , . . . , b m , then we have inv(P ) = inv((a i 1 , a i 2 , . . . , a im )) = |{(k, l); a i k > a i l , k < l}|. 1 ), b 1 ), (σ(a 2  ...  s [pt, i]◮[i+1, q l ] = s [pt, i] s [i+1, q l ] − s [pt, i]↑[i+1, q l ] for k ≤ t ≤ m and 1 ≤ l ≤ k + r −1.  ...

### Proof of Lassalle's Positivity Conjecture on Schur Functions [article]

William Y. C. Chen, Anne X. Y. Ren, Arthur L. B. Yang
2012 arXiv   pre-print
Given a skew partition λ/µ with λ = (λ 1 , λ 2 , . . . , λ ) and µ = (µ 1 , µ 2 , . . . , µ ), let I = (µ + 1, µ −1 + 2, . . . , µ 1 + ), (3.1) J = (λ + 1, λ −1 + 2, . . . , λ 1 + ).  ...  If θ > 0, then we have T (I, J) > 0, since, for 1 ≤ k ≤ , i k = µ +1−k + k ≤ λ +1−k + k = j k . If θ = 0, then we have L = ∞, since f (x) is not a polynomial.  ...

### Melham's Conjecture on Odd Power Sums of Fibonacci Numbers [article]

Brian Y. Sun, Matthew H.Y. Xie, Arthur L. B. Yang
2015 arXiv   pre-print
By the Ozeki-Prodinger formula, we have L 1 L 3 L 5 · · · L 2m+1 n k=1 F 2m+1 2k = L 1 L 3 L 5 · · · L 2m+1 S 2m+1 (F 2n+1 ). Let P 2m−1 (x) = L 1 L 3 L 5 · · · L 2m+1 S 2m+1 /(x − 1) 2 .  ...  = 1, L 0 = 2 and L 1 = 1.  ...

### The Flagged Cauchy Determinant

William Y. C. Chen, Christian Krattenthaler, Arthur L. B. Yang
2005 Graphs and Combinatorics
of [27] ), i.e., for i < j and k < l any path from A i to B l and any path Given the sets A and B of starting and end points, respectively, we may translate an n-tuple (P 1 , P 2 , . . . , P n ) of  ...  For any families L 0 , L 1 , . . . , L n−1 of variables such that |L i | ≤ i, we have s λ (H 1 , H 2 , . . . , H n ) = |h λ j +j−i (H j )| n×n = |h λ j +j−i (H j − L n−i )| n×n , (3. h k (X − Y ) t  ...

### Stable Equivalence over Symmetric Functions

William Y. C. Chen, Arthur L. B. Yang
2005 Electronic Journal of Combinatorics
(b) If L is of Type 2, then we have i ∈ Q Π and i+1 ∈ P Π .  ...  Chen, Yan and Yang [1] obtained the following characterization of outside decompositions of a skew shape.  ...

### Kirillov's Unimodality Conjecture for the Rectangular Narayana Polynomials

Herman Z. Q. Chen, Arthur L. B. Yang, Philip B. Zhang
2018 Electronic Journal of Combinatorics
X l , j > l}|.  ...  . , m is a lattice word of weight (m n ), if the following conditions hold: (a) each i between 1 and m occurs exactly n times and (b) for each 1 r nm and 1 i m − 1, the number of i's in w 1 w 2 · · · w  ...

### Schur Positivity and the q-Log-convexity of the Narayana Polynomials [article]

William Y. C. Chen, Larry X.W. Wang, Arthur L. B. Yang
2008 arXiv   pre-print
For k > l, by (3.13) we have s (2 k ) s (2 l ) = ∆ (2) (s (2 k−1 ) s (2 l ) ), s (2 k+1 ) s (2 l−1 ) = ∆ (2) (s (2 k ) s (2 l−1 ) ).  ...  It follows that s (2 k ) s (2 l ) − s (2 k+1 ) s (2 l−1 ) = ∆ (2) (s (2 k−1 ) s (2 l ) − s (2 k ) s (2 l−1 ) ).  ...
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