A060500 a(n) = number of drops in the n-th permutation of list A060118; the average of digits (where "digits" may eventually obtain also any values > 9) in each siteswap pattern A060496(n).
0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 1, 2, 2, 2, 2, 3, 2, 3, 2
Offset: 0
Keywords
Links
Programs
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Maple
A060500 := avg(Perm2SiteSwap1(PermUnrank3R(n))); # PermUnrank3R(r) gives the permutation with rank r in list A060117: PermUnrank3R := proc(r) local n; n := nops(factorial_base(r)); convert(PermUnrank3Raux(n+1, r, []), 'permlist', 1+(((r+2) mod (r+1))*n)); end; PermUnrank3Raux := proc(n, r, p) local s; if(0 = r) then RETURN(p); else s := floor(r/((n-1)!)); RETURN(PermUnrank3Raux(n-1, r-(s*((n-1)!)), permul(p, [[n, n-s]]))); fi; end; Perm2SiteSwap1 := proc(p) local ip, n, i, a; n := nops(p); ip := convert(invperm(convert(p, 'disjcyc')), 'permlist', n); a := []; for i from 1 to n do a := [op(a), ((ip[i]-i) mod n)]; od; RETURN(a); end; avg := a -> (convert(a,`+`)/nops(a));
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Scheme
(define (A060500 n) (let ((s (+ 1 (A084558 n))) (p (A060118permvec-short n))) (let loop ((d 0) (i 1)) (if (> i s) d (loop (+ d (if (< (vector-ref p (- i 1)) i) 1 0)) (+ 1 i)))))) (define (A060118permvec-short rank) (permute-A060118 (make-initialized-vector (+ 1 (A084558 rank)) 1+) (+ 1 (A084558 rank)) rank)) (define (permute-A060118 elems size permrank) (let ((p (vector-head elems size))) (let unrankA060118 ((r permrank) (i 1)) (cond ((zero? r) p) (else (let* ((j (1+ i)) (m (modulo r j))) (cond ((not (zero? m)) (let ((org-i (vector-ref p i))) (vector-set! p i (vector-ref p (- i m))) (vector-set! p (- i m) org-i)))) (unrankA060118 (/ (- r m) j) j)))))))
Formula
From Antti Karttunen, Aug 18 2016: (Start)
The following formula reflects the original definition of computing the average, with a few unnecessary steps eliminated:
a(n) = 1/s * Sum_{i=1..s} ((i-p[i]) modulo s), where p is the permutation of rank n as ordered in the list A060117, and s is its size (the number of its elements) computed as s = 1+A084558(n).
a(n) = 1/s * Sum_{i=1..s} ((p[i]-i) modulo s). [If inverse permutations from list A060118 are used, then we just flip the order of difference that is used in the first formula].
a(n) = Sum_{i=1..s} [p[i]A060502 for the proof].
(End)
Extensions
Maple code collected together, alternative definition and new formulas added by Antti Karttunen, Aug 24 2016
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