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A129594
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Involution of nonnegative integers induced by the conjugation of the partition encoded in the run lengths of binary expansion of n.
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18
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0, 1, 3, 2, 4, 7, 6, 5, 11, 12, 15, 8, 9, 14, 13, 10, 20, 27, 28, 19, 16, 31, 24, 23, 22, 25, 30, 17, 18, 29, 26, 21, 43, 52, 59, 36, 35, 60, 51, 44, 47, 48, 63, 32, 39, 56, 55, 40, 41, 54, 57, 38, 33, 62, 49, 46, 45, 50, 61, 34, 37, 58, 53, 42, 84, 107, 116, 75, 68, 123
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OFFSET
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0,3
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COMMENTS
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This sequence is based on the fact that compositions (i.e. ordered partitions) can be mapped 1-to-1 to partitions by taking the partial sums of the list where one is subtracted from each composant except the first. (See table A227189 where the parts for each partition are listed).
The inverse process, from partitions to compositions, occurs by inserting the first (i.e. smallest) element of a partition sorted into ascending order to the front of the list obtained by adding one to the first differences of the elements.
Compositions map bijectively to nonnegative integers by assigning each run of k consecutive 1's (or 0's) in binary expansion of n with summand k in the composition.
The graph of this sequence is quite interesting.
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LINKS
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EXAMPLE
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a(8) = 11, as 8 is 1000 in binary, mapping to composition 3+1 (we scan the binary expansion from the least to the most significant end), which maps to partition 3+3, whose conjugate-partition is 2+2+2, yielding composition 2+1+1, which maps to binary 1011, 11 in decimal. a(13) = 14, as 13 is 1101 in binary, mapping to composition 1+1+2, which maps to the partition 1+1+2, whose conjugate-partition is 1+3, yielding composition 1+3, which maps to binary 1110, 14 in decimal. a(11) = 8 and a(14) = 13, as taking the conjugate of a partition is a self-inverse operation.
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PROG
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(MIT/GNU Scheme)
(define (A129594 n) (if (zero? n) n (ascpart_to_binexp (conjugate-partition (binexp_to_ascpart n)))))
(define (conjugate-partition ascpart) (let loop ((conjpart (list)) (ascpart ascpart)) (cond ((null? ascpart) conjpart) (else (loop (cons (length ascpart) conjpart) (delete-matching-items! (map -1+ ascpart) zero?))))))
(define (binexp_to_ascpart n) (let ((runlist (reverse! (binexp->runcount1list n)))) (PARTSUMS (cons (car runlist) (map -1+ (cdr runlist))))))
(define (ascpart_to_binexp ascpart) (runcount1list->binexp (reverse! (cons (car ascpart) (map 1+ (DIFF ascpart))))))
(define (binexp->runcount1list n) (if (zero? n) (list) (let loop ((n n) (rc (list)) (count 0) (prev-bit (modulo n 2))) (if (zero? n) (cons count rc) (if (eq? (modulo n 2) prev-bit) (loop (floor->exact (/ n 2)) rc (1+ count) (modulo n 2)) (loop (floor->exact (/ n 2)) (cons count rc) 1 (modulo n 2)))))))
(define (runcount1list->binexp lista) (let loop ((lista lista) (s 0) (state 1)) (cond ((null? lista) s) (else (loop (cdr lista) (+ (* s (expt 2 (car lista))) (* state (- (expt 2 (car lista)) 1))) (- 1 state))))))
(define (DIFF a) (map - (cdr a) (reverse! (cdr (reverse a)))))
(define (PARTSUMS a) (cdr (reverse! (fold-left (lambda (psums n) (cons (+ n (car psums)) psums)) (list 0) a))))
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CROSSREFS
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Sequences related to partitions encoded in this way:
Cf. A227189 (parts of partitions listed on separate rows of the array).
Cf. A005811 (number of parts in the partition).
Cf. A136480 (for n>= 1, the smallest part).
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KEYWORD
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AUTHOR
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STATUS
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approved
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