• • • • • • • • • • • • • • • • • • • • • • • • Shellable omplex Nonshellable omplex • • • • • • • • X • • • • • • • • 40
• • • • • • • • • • • • • • • • • • • • • • • • Shellable omplex Nonshellable omplex • • • • • • • • X • • • • • • • • 41
• • • • • • • • • • • • • • • • • • • • • • • • Shellable omplex Nonshellable omplex • µ ( σ , τ ) of ∆(( σ , τ )) equals redu ed Euler ha ra teristi • A shellable omplex is homotopi ally a w edge of spheres . • Its redu ed Euler ha ra teristi is the numb er of spheres. • It has nontrivial homology at most in the top dimension. 42
Jason Smith (2014) If σ and τ Theo rem: have the same numb er of des ents, then µ ( σ, τ ) = ( − 1) | τ |−| σ | N ( σ, τ ) , where N ( σ, τ ) of σ in τ is the numb er of no rmal o urren es . Therefo re, | µ ( σ, τ ) | � σ ( τ ) , where σ ( τ ) of σ in τ is the numb er of o urren es . interval [ σ, τ ] And, the is shellable. Pro of: Bije t to sub w o rd o rder and use Bj� rner's results (1988). 43
Jason Smith (2014) If σ and τ Theo rem: have the same numb er of des ents, then µ ( σ, τ ) = ( − 1) | τ |−| σ | N ( σ, τ ) , where N ( σ, τ ) of σ in τ is the numb er of no rmal o urren es . Therefo re, | µ ( σ, τ ) | � σ ( τ ) , where σ ( τ ) of σ in τ is the numb er of o urren es . interval [ σ, τ ] And, the is shellable. Let π Theo rem: b e any p ermutation with a segment of three onse utive numb ers in de reasing o r in reasing o rder. Then µ (1 , π ) = 0 . 44
Jason Smith (2014) If σ and τ Theo rem: have the same numb er of des ents, then µ ( σ, τ ) = ( − 1) | τ |−| σ | N ( σ, τ ) , where N ( σ, τ ) of σ in τ is the numb er of no rmal o urren es . Therefo re, | µ ( σ, τ ) | � σ ( τ ) , where σ ( τ ) of σ in τ is the numb er of o urren es . interval [ σ, τ ] And, the is shellable. Let π Theo rem: b e any p ermutation with a segment of three onse utive numb ers in de reasing o r in reasing o rder. Then µ (1 , π ) = 0 . µ (1 , 71654823) = 0 45
Jason Smith (2014) If σ and τ Theo rem: have the same numb er of des ents, then µ ( σ, τ ) = ( − 1) | τ |−| σ | N ( σ, τ ) , where N ( σ, τ ) of σ in τ is the numb er of no rmal o urren es . Therefo re, | µ ( σ, τ ) | � σ ( τ ) , where σ ( τ ) of σ in τ is the numb er of o urren es . interval [ σ, τ ] And, the is shellable. Let π Theo rem: b e any p ermutation with a segment of three onse utive numb ers in de reasing o r in reasing o rder. Then µ (1 , π ) = 0 . interval [1 , π ] In fa t, the is ontra tible. 46
Jason Smith (2014) If σ and τ Theo rem: have the same numb er of des ents, then µ ( σ, τ ) = ( − 1) | τ |−| σ | N ( σ, τ ) , where N ( σ, τ ) of σ in τ is the numb er of no rmal o urren es . Therefo re, | µ ( σ, τ ) | � σ ( τ ) , where σ ( τ ) of σ in τ is the numb er of o urren es . interval [ σ, τ ] And, the is shellable. There a re results/ onje tures analogous to the ab ove fo r the la y ered and sepa rable p ermutations. 47
Sagan-V atter (2005): M�bius fun tion fo r la y ered p ermutations 48
Sagan-V atter (2005): M�bius fun tion fo r la y ered p ermutations 3 2 1 5 4 6 8 7 49
Sagan-V atter (2005): M�bius fun tion fo r la y ered p ermutations 3 2 1 5 4 6 8 7 50
Sagan-V atter (2005): M�bius fun tion fo r la y ered p ermutations 3 2 1 5 4 6 8 7 A la y ered p ermutation is a on atenation of de reasing se- quen es, ea h smaller than the next. 51
Sagan-V atter (2005): M�bius fun tion fo r la y ered p ermutations • • • • • • • • 3 2 1 5 4 6 8 7 A la y ered p ermutation is a on atenation of de reasing se- quen es, ea h smaller than the next. 52
Sagan-V atter (2005): M�bius fun tion fo r la y ered p ermutations • • • • • • • • 3 2 1 5 4 6 8 7 53
Sagan-V atter (2005): M�bius fun tion fo r la y ered p ermutations • • • • • • • • 3 2 1 5 4 6 8 7 (Any subsequen e of a la y ered p ermutation is la y ered) 54
Sagan-V atter (2005): M�bius fun tion fo r la y ered p ermutations • • • • • • • • 3 2 1 5 4 6 8 7 An e�e tive fo rmula, but to o long to �t inside these ma rgins . . . 55
Sagan-V atter (2005): M�bius fun tion fo r la y ered p ermutations • • • • • • • • 3 2 1 5 4 6 8 7 An e�e tive fo rmula, but to o long to �t inside these ma rgins . . . (Simila r to p ermutations with �xed numb er of des ents) 56
Sagan-V atter (2005): M�bius fun tion fo r la y ered p ermutations • • • • • • • • 3 2 1 5 4 6 8 7 A sp e ial ase of the sepa rable p ermutations. 57
• • • • • • • • 4 2 3 5 1 7 8 6 58
• • • • • • • • 4 2 3 5 1 7 8 6 59
• • • • • • • • 4 2 3 5 1 7 8 6 A de omp osable p ermutation is a dire t sum 42351786 = 42351 ⊕ 231 60
• • • • • • • • • • • • • • • • 4 2 3 5 1 7 8 6 A de omp osable p ermutation A sk ew-de omp osable is a dire t sum p ermutation is a sk ew sum 42351786 = 42351 ⊕ 231 76841325 = 213 ⊖ 41325 61
• A p ermutation is sepa rable if • it an b e generated from 1 b y • dire t sums and sk ew sums. • • • • • 4 2 3 5 1 7 8 6 62
• A p ermutation is sepa rable if • it an b e generated from 1 b y • dire t sums and sk ew sums. • • • • • 4 2 3 5 1 7 8 6 63
• A p ermutation is sepa rable if • it an b e generated from 1 b y • dire t sums and sk ew sums. • • • • • 4 2 3 5 1 7 8 6 64
• A p ermutation is sepa rable if • it an b e generated from 1 b y • dire t sums and sk ew sums. • • • • • 4 2 3 5 1 7 8 6 65
• A p ermutation is sepa rable if • it an b e generated from 1 b y • dire t sums and sk ew sums. • • • • • 4 2 3 5 1 7 8 6 66
• A p ermutation is sepa rable if • it an b e generated from 1 b y • dire t sums and sk ew sums. • • • • • 4 2 3 5 1 7 8 6 67
• A p ermutation is sepa rable if • it an b e generated from 1 b y • dire t sums and sk ew sums. • • • • • 4 2 3 5 1 7 8 6 68
• A p ermutation is sepa rable if • it an b e generated from 1 b y • dire t sums and sk ew sums. • • • • • 4 2 3 5 1 7 8 6 Sepa rable 69
• A p ermutation is sepa rable if • it an b e generated from 1 b y • dire t sums and sk ew sums. • • De omp oses b y sk ew/dire t • sums into singletons • • 4 2 3 5 1 7 8 6 Sepa rable 70
• A p ermutation is sepa rable if • it an b e generated from 1 b y • dire t sums and sk ew sums. • • De omp oses b y sk ew/dire t • sums into singletons • • 4 2 3 5 1 7 8 6 Sepa rable A p ermutation is sepa rable if and only if it avoids the pat- terns 2413 and 3142. 71
• A p ermutation is sepa rable if • it an b e generated from 1 b y • dire t sums and sk ew sums. • • De omp oses b y sk ew/dire t • sums into singletons • • 4 2 3 5 1 7 8 6 Sepa rable A p ermutation is sepa rable if • and only if it avoids the pat- • terns 2413 and 3142. • • 2 4 1 3 Not sepa rable 72
Burstein, Jel�nek, Jel�nk ov�, Steingr�msson: If σ and τ Theo rem: a re sepa rable p ermutations, then � µ ( σ , τ ) = � 1) parity(X) ( X ∈OP of σ in τ where the sum is over unpaired o urren es . 73
Burstein, Jel�nek, Jel�nk ov�, Steingr�msson: If σ and τ Theo rem: a re sepa rable p ermutations, then � µ ( σ , τ ) = � 1) parity(X) ( X ∈OP of σ in τ where the sum is over unpaired o urren es . 74
Li − → Nan L Sagan − → Bru e S 75
Li − → Nan L Sagan − → Bru e S E. Babson, A. Bj� rner, L, V. W elk er, J. Sha reshian 76
Li − → Nan L Sagan − → Bru e S 77
Li − → Nan L Sagan − → Bru e S Billera − → Lou B 78
Li − → Nan L Sagan − → Bru e S Billera − → Lou B a hs − → Mi helle W MW 79
Li − → Nan L Sagan − → Bru e S Billera − → Lou B a hs − → Mi helle W MW ra − → P eter M Nama M N 80
Li − → Nan L Sagan − → Bru e S Billera − → Lou B a hs − → Mi helle W MW ra − → P eter M Nama M N Abb reviating y our last name to a single letter implies every- b o dy should rememb er y our name. 81
Li − → Nan L Sagan − → Bru e S Billera − → Lou B a hs − → Mi helle W MW ra − → P eter M Nama M N Abb reviating y our last name to a single letter implies every- b o dy should rememb er y our name. Let's put an end to this immo dest y! 82
Li − → Nan L Sagan − → Bru e S Billera − → Lou B a hs − → Mi helle W MW ra − → P eter M Nama M N Abb reviating y our last name to a single letter implies every- b o dy should rememb er y our name. Let's put an end to this immo dest y! (Unless y our name is Central Ship y a rd, in whi h ase y ou ma y b e fo rgiven) 83
Burstein, Jel�nek, Jel�nk ov�, Steingr�msson: If σ and τ Theo rem: a re sepa rable p ermutations, then � µ ( σ , τ ) = � 1) parity(X) ( X ∈OP of σ in τ where the sum is over unpaired o urren es . 84
Burstein, Jel�nek, Jel�nk ov�, Steingr�msson: If σ and τ Theo rem: a re sepa rable p ermutations, then � µ ( σ , τ ) = � 1) parity(X) ( X ∈OP of σ in τ where the sum is over unpaired o urren es . omputes µ ( σ , τ ) (This in p olynomial time) 85
Burstein, Jel�nek, Jel�nk ov�, Steingr�msson: If σ and τ Theo rem: a re sepa rable p ermutations, then � µ ( σ , τ ) = � 1) parity(X) ( X ∈OP of σ in τ where the sum is over unpaired o urren es . If σ and τ Co rolla ry: a re sepa rable then | µ ( σ , τ ) | � σ ( τ ) where σ ( τ ) of σ in τ is the numb er of o urren es . 86
Burstein, Jel�nek, Jel�nk ov�, Steingr�msson: If σ and τ Theo rem: a re sepa rable p ermutations, then � µ ( σ , τ ) = � 1) parity(X) ( X ∈OP of σ in τ where the sum is over unpaired o urren es . If σ and τ Co rolla ry: a re sepa rable then | µ ( σ , τ ) | � σ ( τ ) where σ ( τ ) of σ in τ is the numb er of o urren es . (A generalization of a onje ture of T enner and Steingr�ms- son) 87
Burstein, Jel�nek, Jel�nk ov�, Steingr�msson: If σ and τ Theo rem: a re sepa rable p ermutations, then � µ ( σ , τ ) = � 1) parity(X) ( X ∈OP of σ in τ where the sum is over unpaired o urren es . If σ and τ Co rolla ry: a re sepa rable then | µ ( σ , τ ) | � σ ( τ ) where σ ( τ ) of σ in τ is the numb er of o urren es . � n + k − 1 � µ (135 . . . (2 k - 1) (2 k ) . . . 42 , 135 . . . (2 n -1 ) (2 n ) . . . 42) = n − k 88
Burstein, Jel�nek, Jel�nk ov�, Steingr�msson: If σ and τ Theo rem: a re sepa rable p ermutations, then � µ ( σ , τ ) = � 1) parity(X) ( X ∈OP of σ in τ where the sum is over unpaired o urren es . If σ and τ Co rolla ry: a re sepa rable then | µ ( σ , τ ) | � σ ( τ ) where σ ( τ ) of σ in τ is the numb er of o urren es . � 8 / 2+4 / 2 − 1 � � 5 � µ (1342 , 13578642) = = 8 / 2 − 4 / 2 2 89
Burstein, Jel�nek, Jel�nk ov�, Steingr�msson: If σ and τ Theo rem: a re sepa rable p ermutations, then � µ ( σ , τ ) = � 1) parity(X) ( X ∈OP of σ in τ where the sum is over unpaired o urren es . If σ and τ Co rolla ry: a re sepa rable then | µ ( σ , τ ) | � σ ( τ ) where σ ( τ ) of σ in τ is the numb er of o urren es . then µ ( If τ 1 , τ ) ∈ { 0 , 1 , − 1 } . Co rolla ry: is sepa rable, 90
Burstein, Jel�nek, Jel�nk ov�, Steingr�msson: If σ and τ Theo rem: a re sepa rable p ermutations, then � µ ( σ , τ ) = � 1) parity(X) ( X ∈OP of σ in τ where the sum is over unpaired o urren es . If σ and τ Co rolla ry: a re sepa rable then | µ ( σ , τ ) | � σ ( τ ) where σ ( τ ) of σ in τ is the numb er of o urren es . then µ ( If τ 1 , τ ) ∈ { 0 , 1 , − 1 } . Co rolla ry: is sepa rable, Neither o rolla ry true in general 91
La y ered intervals and �xed-des intervals a re isomo rphi to t w o extremes in the Generalized sub w o rd o rder (determined oset P b y a p ) of Sagan and V atter. 92
La y ered intervals and �xed-des intervals a re isomo rphi to t w o extremes in the Generalized sub w o rd o rder (determined oset P b y a p ) of Sagan and V atter. P anti hain: · · · 1 2 3 4 • • • • · · · 1344 � P 113414 1343 � � P 113414 93
La y ered intervals and �xed-des intervals a re isomo rphi to t w o extremes in the Generalized sub w o rd o rder (determined oset P b y a p ) of Sagan and V atter. P anti hain: · · · 1 2 3 4 • • • • · · · 1344 � P 113414 1343 � � P 113414 94
La y ered intervals and �xed-des intervals a re isomo rphi to t w o extremes in the Generalized sub w o rd o rder (determined oset P b y a p ) of Sagan and V atter. P P hain: anti hain: . . · · · 1 2 3 4 . • 4 • • • • · · · • 3 • 2 1344 � P 113414 • 1 1343 � � P 113414 1343 � P 113414 95
La y ered intervals and �xed-des intervals a re isomo rphi to t w o extremes in the Generalized sub w o rd o rder (determined oset P b y a p ) of Sagan and V atter. P P hain: anti hain: . . · · · 1 2 3 4 . • 4 • • • • · · · • 3 • 2 1344 � P 113414 • 1 ( 3 < P 4 1343 � � P 113414 1343 � P 113414 ) 96
La y ered intervals and �xed-des intervals a re isomo rphi to t w o extremes in the Generalized sub w o rd o rder (determined oset P b y a p ) of Sagan and V atter. P P hain: anti hain: . . · · · 1 2 3 4 . • 4 • • • • · · · • 3 • 2 1344 � P 113414 • 1 1343 � � P 113414 1343 � P 113414 Fixed-des La y ered 97
La y ered intervals and �xed-des intervals a re isomo rphi to t w o extremes in the Generalized sub w o rd o rder (determined oset P b y a p ) of Sagan and V atter. P P hain: anti hain: . . · · · 1 2 3 4 . • 4 • • • • · · · • 3 • 2 1344 � P 113414 • 1 1343 � � P 113414 1343 � P 113414 Fixed-des La y ered Is there a family of intervals of p ermutations interp olating b et w een these t w o extremes (that a re shellable, o r at least with a tra table M�bius fun tion)? 98
M Nama ra-Steingr�msson (refo rmulation of BJJS): Let τ = τ 1 ⊕ · · · ⊕ τ k Theo rem: b e �nest de omp osition. Then � � µ ( σ , τ ) = µ ( σ m , τ m ) + ǫ m σ = σ 1 ⊕ ... ⊕ σ k m if σ m = ∅ and τ m − 1 = τ m 1 , ǫ m = where 0 , else 99
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