UNIFORMS THAT WORK FOR YOU039a82463b8d5029219d-c0812be71c009a193aa9dc03fe24234a.r86… · Weight:...
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TRIPLE FACTORIZATION OF SOME RIORDAN MATRICES Paul Peart* Department of Mathematics, Howard University, Washington, D.C. 20059 Leon Woodson Department of Mathematicss, Morgan State University, Baltimore, MD 21239 (Submitted June 1991) 1. INTRODUCTION When examining a combinatorial sequence, generating functions are often useful. That is, if we are interested in analyzing the sequence a 0 , a l? a 2 ,..., we investigate the formal power series f(x) = a 0 +a l x + a 2 x 2 -\— . In a recent paper [2], techniques are discussed that assist in finding closed-form expressions for the formal power series for a select, but large, set of combinatorial sequences. The methods involve using infinite matrices and the Riordan group. The Riordan group is defined in section 2 of this paper. Each matrix, L, in the Riordan group is associated with a combinatorial sequence and with a matrix, S L , called the Stieltjes matrix. S L is defined in section 3. In this paper, we show that when S L is tridiagonal, then L= PCF, where the first factor P is a Pascal-type matrix, the second factor C involves the generating function for the Catalan numbers, and the third factor F involves the Fibonacci generating function. The following is an example: © 1 1 .I 11 1 69 13 1 . "1 1 1 o • 12 1 v . 13 3 1 14 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 . r i 0 -i 0 2 0 -5 1 0 -2 0 5 0 1 0 -3 0 9 1 0 -4 0 o : 1 0 1 -501. r i 11 0 • 2 11 3 2 11 5 3 2 11 8 5 3 2 11 13 8 5 3 2 1 1 . The matrices in the Riordan group are infinite and lower triangular. So the example shows only the first seven rows. The first factor on the right is the Pascal matrix. The first column in the second factor has C(-x 2 ) as generating function, where is the generating function for the Catalan numbers. The third factor has the Fibonacci numbers in each column. See section 6 for further examples of this triple factorization. * Partially supported by NSF grant 634012. 1 3 1 9 5 1 27 18 7 1 81 56 31 9 243 162 109 48 729 458 332 194 1993] 121
Transcript of UNIFORMS THAT WORK FOR YOU039a82463b8d5029219d-c0812be71c009a193aa9dc03fe24234a.r86… · Weight:...
2020 COLLECTION
LONG AND SHORT SLEEVE STYLES
NEW! ZIP FASTENING JACKETS
Page 4–17 Page 5
JOGGER & CARGO CHEF’S TROUSERS
Pages 19–20
UNIFORMS THAT WORK FOR YOU
cONTENTSChef’s Jackets 4 - 17
Chef’s T-Shirts 18
Chef’s Trousers 19 - 21
Chef’s Accessories 22 - 23
Chef’s Aprons 24 - 27
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white
black denim
indigo denim
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UNIFORMS THAT WORK FOR YOU
zipped chef’s jacketsNEW
Colour contrast underarm
Zip through fastening
Loop for hanging
PR906
Chef’s Zip-Close Short Sleeve Jacket Zip through fastening Double pen pocket on left sleeve Colour contrast underarm Mandarin collar
FABRICWhite: 65% Polyester, 35% CottonDenim: 60% Cotton, 40% Polyester Weight: 170gsm/230gsm Denim
SIZESXS/30” – 2XL/50”
COLOUR3 as listed
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PR901PR900
white white
black black
UNIFORMS THAT WORK FOR YOU
PR900
‘essential’ short sleeve chef’s jacket Five button wrap over fastening Long life resin buttons Double pen pocket on left sleeve Mandarin collar
FABRIC65% Polyester, 35% CottonWeight: 170gsm
SIZESXS/30” – 4XL/58”
COLOUR2 as listed
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PR901
‘essential’ long sleeve chef’s jacket Five button wrap over fastening Long life resin buttons Double pen pocket on left sleeve Mandarin collar
FABRIC65% Polyester, 35% CottonWeight: 170gsm
SIZESXS/30” – 4XL/58”
COLOUR2 as listed
Long life resin buttons
‘essential’ chef’s jacketsNEW
6 7
white
steel
PR656
PR657
black
black
steel
white
PR656
Short Sleeve Chef’s Jacket Ten button hole wrap over fastening Double pen pocket on left sleeve Mandarin collar
FABRIC65% Polyester, 35% Cotton Weight: 195gsm
SIZESXS/30” – 4XL/58”
COLOUR3 as listed
PR657
Long Sleeve Chef’s Jacket Ten button hole wrap over fastening Double pen pocket on left sleeve Mandarin collar
FABRIC65% Polyester, 35% Cotton Weight: 195gsm
SIZESXS/30” – 4XL/58”
COLOUR3 as listed
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CLASSIC CHEF’S JACKETS
UNIFORMS THAT WORK FOR YOU
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white
white
PR670PR671
PR670
Short Sleeve Chef’s JacketPR671
long Sleeve Chef’s Jacket Ten button hole wrap over fastening Double pen pocket on left sleeve Back panels for a more fitted style
FABRIC65% Polyester, 35% Cotton Weight: 195gsm
SIZES8/XS - 18/2XL
COLOUR1 as listed
women’s STYLES
1110
white
black
Drawing moisture away from the body.
Allows air flow through the fabric.
Fast dry and crease resistant.
coolchecker® fabric technology
PR903
chef’s long sleeve coolchecker® jacket Coolchecker® back panel Double pen pocket on left sleeve Self-fabric ten button fastening
FABRIC65% Polyester, 35% Cotton Weight: 170gsm
SIZESXS/30” – 2XL/50”
COLOUR2 as listed
NEWcoolchecker® long sleeve chef’s jacket
Coolchecker® fabric back panel and loop for hanging
1312
whiteblack
whiteblack
PR664
PR665
UNIFORMS THAT WORK FOR YOU
PRESS-STUD FASTENINGPR664
Short Sleeve Studded Front Chef’s JacketPR665
Long Sleeve Studded Front Chef’s Jacket Mandarin collar Ten press-stud fastening Wrap over style
FABRIC65% Polyester, 35% Cotton Weight: 195gsm
SIZESXS/30” – 3XL/54”
COLOUR2 as listed
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1514
black denim
black
white
PR660 PR661
grey denim
PR661
Long Sleeve ‘Cuisine’ Chef’s Jacket Mandarin collar Comes with Black removable studs
(other colours available, see page 23)
Ten hole wrap over fastening
FABRIC65% Polyester, 35% Cotton Weight: 195gsm
SIZESXS/30” – 4XL/58”
COLOUR2 as listed
PR660
DENIM CHEF’S JACKET Cotton rich chef’s jacket Cross-over front fastening Ten press-stud fastening
FABRIC60% Cotton, 40% Polyester Weight: 230gsm
SIZESXS/30” – 3XL/54”
COLOUR2 as listed
denimCHEF’S JACKET
‘CUISINE’CHEF’S JACKET
COTTON RICH ˚
1716
black denim
indigo denim
black
white
MODERNFIT
PR556
chef’s ‘artisan’ jogger bottoms Cuffed ankle Elasticated waistband, draw
cord and belt loops Soft and stretchy fabric
FABRIC87% Cotton, 12% Polyester, 1% Elastane Weight: 230gsm
SIZESXS/28” – 2XL/44”
COLOUR2 as listed
NEWPR649
Chef’s Coolchecker® T-shirt Crew neck styling Coolchecker® fabric back panel Underarm vent holes Pen pocket on left sleeve
FABRIC100% Polyester Weight: 135gsm
SIZESXS/30” – 3XL/54”
COLOUR2 as listed
Underarm vent holes
UNIFORMS THAT WORK FOR YOU
CHEF’ST-SHIRT
CHEF’STROUSERS
COTTON RICH
1918
black navy & white check black & white check
black & white check black
NEW
PR555
chef’s ‘essential’ cargo pocket trouser Two mobile friendly cargo pockets Elasticated waistband Draw cord
FABRIC65% Polyester, 35% Cotton Weight: 195gsm
SIZESXS/28” – 3XL/46”
COLOURBlack
Draw cord waist
PR554
Chef’s ‘Select’ Slim Leg Trouser Fitted style on lower leg Patch pocket on right leg Elasticated waist with draw cord
and zip fly fastening
FABRIC65% Polyester, 35% Cotton Weight: 195gsm
SIZESXS/28” – 2XL/44”
COLOUR1 as listed
PR552
Pull on Chef’s Trouser Elasticated waist Zip fly 77cm/30.5” inside leg
FABRIC65% Polyester, 35% Cotton Weight: 195gsm
SIZESXS/28” – 2XL/44”
COLOUR2 as listed
PR553
‘Essential’ Chef’s Trouser Elasticated waist Swing pockets 77cm/30.5” inside leg
FABRIC65% Polyester, 35% Cotton Weight: 195gsm
SIZESXS/28” – *4XL/48” *only available in Black
COLOUR2 as listed
chef’s trousers
2120
NAVYBLACK WHITE
PR658
PR652
PR648
black
indigo denim†
navy bottle white red
black denim† grey denim† black/white check
PR653
white
white
black
black
PR652
Chef’s Jacket Studs Removable studs For use with PR661 Cuisine Jacket (pg 17) Sold in bags of 12
FABRIC COLOURDurable plastic 3 as listed Wipe clean
PR658
Chef’s Zandana Two ties at the back Elasticated back Neck flap
FABRIC65% Polyester, 35% Cotton. Weight: 195gsm
COLOUR2 as listed
PR648
Turn up Chef’s Hat Turn up double edge band Five eyelets for ventilation Four sizes available
FABRIC65% Polyester, 35% Cotton. Weight: 195gsm
SIZESS 54cm, M 57cm, L 60cm, XL 63cm
COLOUR2 as listed
Elasticated back
NEW COLOUR
NEW COLOUR
PR653
Chef’s Skull Cap Flat top Elasticated back One size fits all
FABRICColours: 65% Polyester, 35% Cotton, 195gsm Denim: 60% Cotton, 40% Polyester, 230gsm
COLOUR9 as listed
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COTTON RICH
chef’s accessories
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cornflower royal marine blue navy indigo denim blue denim
cyan light blue peacock mid blue sapphire lilac
lavender purple rich violet aubergine fuchsia magenta
pink hot pink raspberry crush burgundy red salsa
strawberry red orange terracotta chestnut brown mocha
mushroom latte khaki natural camel mustard
sunflower yellow lemon lime oasis green olive
aqua duck egg blue turquoise teal emerald kelly
pistachio apple bottle sage white* black denim
grey denim silver steel charcoal dark grey black*
PR150
‘colours collection’ bib apron
PR150
Adjustable buckle on neckband Self-fabric ties Two sizes available
FABRICColours: 65% Polyester, 35% Cotton, 195gsm Denim: 70% Cotton, 30% Polyester, 265gsm
SIZESStandard size: 86cm x 72cm * Extra large size: 102cm x 91cm Black and White only
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Colours
2524
ORANGE
KHAKI
NATURAL
BROWN
BOTTLE
APPLE
LIME
NAVY
TURQUOISE
ROYAL
black/white † navy/white†black/grey
STEEL
WHITE
SILVER
DARK GREY
BLACK
PR110
RED
BURGUNDY
HOT PINK
AUBERGINE
PURPLE
PR102
20Colours
UNIFORMS THAT WORK FOR YOU
PR102
100% COTTON Bib Apron Self-fabric ties Self-fabric neckband Width 60cm, length 87cm
FABRIC100% Cotton Weight 240gsm
COLOUR20 as listed
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100% COTTON
PR110
Striped Bib Apron Butcher’s style apron 90cm self-fabric ties and neck band Width 72cm, length 101cm
FABRICNavy/White & Black/White: 100% Cotton, 240gsm Black/Grey: 65% Polyester, 35% Cotton, 195gsm
COLOUR3 as listed
100% COTTON
†
100% cotton
2726
UNIFORMS THAT WORK FOR YOU
JacketsT-SHIRTStrousersaccessoriesaprons
