AlternatingSignMatrix
A name from the enumeratio catalogue with no kernel of its own yet; this page carries its crosswalk only.
Catalogued in the enumeratio database; not yet written up here.
Wikipedia
Alternating sign matrixvia AlternatingSignMatricesOEISA005130via AlternatingSignMatricesSageMathAlternatingSignMatrices(n)via AlternatingSignMatricesAlternatingSignMatrix(...)Catalogued in the enumeratio database; not yet written up here.Typed signatures
| Type | Package | Applies |
|---|---|---|
(list<list<integer>>) -> alternating_sign_matrix | combinatorics | always |
Statistics
On AlternatingSignMatrix, called as CombinatorialStat(x, "Name").
- AlternatingSignMatrixPosetRankSt000076Half the sum of (i - j)^2 a(i, j): the rank in the alternating sign matrix poset.
- DecreasableCornerSumPositionsSt001965The interior positions of the corner sum matrix whose entry can be lowered by one to give another alternating sign matrix's corner sum matrix.
- DistinctDiagonalSumCountSt000893The number of distinct sums of the entries along a diagonal.
- FirstColumnOnePositionSt000193The row of the unique 1 in the first column.
- FirstRowOnePositionSt000066The column of the unique 1 in the first row.
- LastColumnOnePositionSt000200The row of the unique 1 in the last column.
- LastRowOnePositionSt000199The column of the unique 1 in the last row.
- MainDiagonalSumSt000894The sum of the entries on the main diagonal, its trace.
- MatrixInversionNumberSt000067The sum of a(i, j) a(k, l) over the pairs of entries with i > k and j < l.
- MatrixPermanentSt001260The permanent: the sum over the permutations s of the product of a(i, s(i)).
- MaximalDiagonalNonzeroCountSt000892The largest number of nonzero entries along a diagonal.
- MaximalDiagonalSumSt000888The largest sum of the entries along a diagonal.
- MinimalRowZeroCountSt001925The fewest zeros in a row.
- NegativeOneCountSt000065The number of entries equal to -1.
- NonzeroEntryCountSt000890The number of nonzero entries.
- OnesOnMainDiagonalSt000895The number of 1s on the main diagonal.
- PositiveMatrixInversionNumberSt000332The inversion number less the number of entries equal to -1.
- PositiveOneCountSt000197The number of entries equal to 1.
- TopLeftZeroSquareSizeSt001981The side of the largest square of zeros at the top left corner.
- ZerosOnMainDiagonalSt000896The number of 0s on the main diagonal.
Maps
Into and out of AlternatingSignMatrix, called as CombinatorialMap(x, "Name"). Properties are the laws we state and what FindStat records.
13 more in FindStat, not implemented here
- Mp00001AlternatingSignMatrix → SemistandardTableauInjectionGraded mapto semistandard tableau via monotone triangles
- Mp00002AlternatingSignMatrix → PermutationSurjectionGraded mapto left key permutation
- Mp00003AlternatingSignMatrix → AlternatingSignMatrixBijectionGraded maprotate counterclockwise
- Mp00004AlternatingSignMatrix → AlternatingSignMatrixBijectionGraded maprotate clockwise
- Mp00005AlternatingSignMatrix → AlternatingSignMatrixBijectionGraded mapInvolutiontranspose
- Mp00006AlternatingSignMatrix → AlternatingSignMatrixBijectionGraded mapgyration
- Mp00007AlternatingSignMatrix → DyckPathSurjectionGraded mapto Dyck path
- Mp00035DyckPath → AlternatingSignMatrixInjectionGraded mapto alternating sign matrix
- Mp00063Permutation → AlternatingSignMatrixInjectionGraded mapto alternating sign matrix
- Mp00098AlternatingSignMatrix → PerfectMatchingGraded maplink pattern
- Mp00137DyckPath → AlternatingSignMatrixInjectionGraded mapto symmetric ASM
- Mp00333AlternatingSignMatrix → DyckPathSurjectionGraded maplink pattern
- Mp00334AlternatingSignMatrix → DyckPathSurjectionGraded mapdiagonal height