DescentTopWord from Permutation

Letter v - 1 is 1 when value v is a descent top: some smaller value immediately follows it.

Permutation → BinaryWordsurjection

Value 1 is never a descent top, so the first letter stands for 2.

Called as CombinatorialMap(x, "DescentTopWord"). The head's page has its examples.

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Each element of Permutation at a size, and its image, applied in your browser; sizes past 7 are not offered.

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Sizes 1 to

Transport a statistic

Pick a statistic on BinaryWord. Its distribution over the images of Permutation at each size is computed, and compared with the 323 statistics on Permutation: those that share it are the ones the map sends it to, as far as the sizes computed show.

Statistic HammingWeightLeadingOnesCountOnePositionSumOneRunCountOddOneRunCountLongestRunOfOnesChangeCountPathDepthOnesMinusZerosBalancedPrefixCountSymmetryDegreeAscentsInversionsMajorIndexLongestConstantRunLongestAlternatingFactorFirstOnePositionAfterAppendedOneDecimalRepresentationDecimalRepresentationWithLeadingOneDistinctFactorCountBorderLengthSymmetricBorderLengthInversionSumNonInversionSumStrictlyIncreasingRunsMinimalPeriodBorderCountWeakMajorIndexDistinctSubsequenceCountRepetitionExponentBalanceShortestPalindromicDecompositionLengthPalindromicDecompositionCountLongestCatalanFactorSemilengthCatalanFactorCountCriticalStepCountInternalInversionsUniqueFactorLengthLyndonFactorizationLengthLongestYamanouchiPrefixLongestPalindromicPrefixLongestPalindromicFactorLongestPalindromicSubsequenceDistinctCubeCountDistinctSquareCountCyclicShiftIndexDescentWordPreimageCountLatticePathsAboveWordCountSameEndpointPathsAboveWordCountEvenPalindromicPrefixHalfLengthOddPalindromicPrefixHalfLengthLongestPalindromicFactorStartingWithOneReverseComplementFactorHalfLengthReverseComplementFactorHalfLengthStartingWithOneFlexModularMajorIndexValleyFillingDegreeBaselineCrossingsPrimitiveFactorCount Sizes 1 to

FindStat Mp00130

descent tops

Matched by value: our map agrees with FindStat's image on each of the 873 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

SurjectionGraded mapas FindStat records them, checked by FindStat's contributors.

The descent tops of a permutation as a binary word. Since 1 is never a descent top, it is omitted and the first letter of the word corresponds to the element 2.

Object	Image
[1]
[1,2]	0
[2,1]	1
[1,2,3]	00

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00130 has the rest.