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Division and the Giambelli Identity
[article]

2005
*
arXiv
*
pre-print

{

arXiv:math/0504487v1
fatcat:wrvzphj7gndltoex3eiybp6qnq
*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,*b*∈*B*(a −*b*), and by A −*B*the set difference. ...##
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Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences
[article]

2010
*
arXiv
*
pre-print

≤ 1. (2.2) In order to show that (x + 1)

arXiv:1007.5017v1
fatcat:wx7ysqdfbfasjmjdkqxjxhonsi
*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 ...##
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The srank Conjecture on Schur's Q-Functions
[article]

2008
*
arXiv
*
pre-print

Let Q (a) = q a and Q (a,

arXiv:0805.2782v1
fatcat:4xvaxxlsqfemtg6udonrtpjgy4
*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*B*−*B*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 . ...##
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The Zrank Conjecture and Restricted Cauchy Matrices
[article]

2005
*
arXiv
*
pre-print

The length

arXiv:math/0504488v1
fatcat:r2bawsv55jeurnezfhfmt2dkxi
*ℓ*(S) of a snake S is defined to be one less than its number of squares. For an empty snake S, let*ℓ*(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. ...##
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Branden's Conjectures on the Boros-Moll Polynomials
[article]

2012
*
arXiv
*
pre-print

Define

arXiv:1205.0305v1
fatcat:rmijxqa5qbaxzezefsrmdqblri
*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. ...##
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On the Modes of Polynomials Derived from Nondecreasing Sequences
[article]

2010
*
arXiv
*
pre-print

Chen,

arXiv:1008.4927v1
fatcat:cjg263zrxjcchoj6lhm43duaei
*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). ...##
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A Blass-Sagan bijection on Eulerian equivalence classes
[article]

2007
*
arXiv
*
pre-print

By definitions, the two orientations (

arXiv:0706.3263v1
fatcat:3xkxpsbiyfhi3llch36fmw2poe
*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. ...##
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Stable Equivalences of Giambelli Type Matrices of Schur Functions
[article]

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

arXiv:math/0508008v1
fatcat:p7tyrztusze45ovhhmmqlkvesi
*L*(*b*) If ... Chen, Yan and*Yang*[1] obtained the following characterization of outside decompositions of a skew shape. ...##
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Transformations of Border Strips and Schur Function Determinants
[article]

2004
*
arXiv
*
pre-print

Suppose

arXiv:math/0406250v1
fatcat:aycoq32aeve4voc3hbuiy7k3vu
*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. ...##
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Proof of Lassalle's Positivity Conjecture on Schur Functions
[article]

2012
*
arXiv
*
pre-print

Given a skew partition λ/µ with λ = (λ 1 , λ 2 , . . . , λ

arXiv:1209.0078v1
fatcat:pl7kwvpcozhljet3bzw4axxjfq
*ℓ*) and µ = (µ 1 , µ 2 , . . . , µ*ℓ*), let I = (µ*ℓ*+ 1, µ*ℓ*−1 + 2, . . . , µ 1 +*ℓ*), (3.1) J = (λ*ℓ*+ 1, λ*ℓ*−1 + 2, . . . , λ 1 +*ℓ*). ... If θ > 0, then we have T (I, J) > 0, since, for 1 ≤ k ≤*ℓ*, i k = µ*ℓ*+1−k + k ≤ λ*ℓ*+1−k + k = j k . If θ = 0, then we have*L*= ∞, since f (x) is not a polynomial. ...##
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Melham's Conjecture on Odd Power Sums of Fibonacci Numbers
[article]

2015
*
arXiv
*
pre-print

By the Ozeki-Prodinger formula, we have

arXiv:1502.03294v1
fatcat:5mhwkp5sobhjbhe5uikclksbpu
*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. ...##
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The Flagged Cauchy Determinant

2005
*
Graphs and Combinatorics
*

of [27] ), i.e., for i < j and k <

doi:10.1007/s00373-004-0593-9
fatcat:kodlfryqyfcejdpmm2v3gyok3a
*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 ...##
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Stable Equivalence over Symmetric Functions

2005
*
Electronic Journal of Combinatorics
*

(

doi:10.37236/1880
fatcat:ng7uu7wqcrhzvhn2isrsye3acy
*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. ...##
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Kirillov's Unimodality Conjecture for the Rectangular Narayana Polynomials

2018
*
Electronic Journal of Combinatorics
*

X

doi:10.37236/6806
fatcat:a5e7ntvdtzfhrka2pdigspd2ru
*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 ...##
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Schur Positivity and the q-Log-convexity of the Narayana Polynomials
[article]

2008
*
arXiv
*
pre-print

For k >

arXiv:0806.1561v1
fatcat:5wggi267wfbhfl2b2zy7mtnomy
*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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