There are 403 statistics
on Permutations
in the database:
(and possibly some waiting for verification)
(and possibly some waiting for verification)
St000001Permutations ⟶ ℤ
The number of reduced words for a permutation.
St000002Permutations ⟶ ℤ
The number of occurrences of the pattern 123 in a permutation.
St000004Permutations ⟶ ℤ
The major index of a permutation.
St000007Permutations ⟶ ℤ
The number of saliances of the permutation.
St000018Permutations ⟶ ℤ
The number of inversions of a permutation.
St000019Permutations ⟶ ℤ
The cardinality of the support of a permutation.
St000020Permutations ⟶ ℤ
The rank of the permutation.
St000021Permutations ⟶ ℤ
The number of descents of a permutation.
St000022Permutations ⟶ ℤ
The number of fixed points of a permutation.
St000023Permutations ⟶ ℤ
The number of inner peaks of a permutation.
St000028Permutations ⟶ ℤ
The number of stack-sorts needed to sort a permutation.
St000029Permutations ⟶ ℤ
The depth of a permutation.
St000030Permutations ⟶ ℤ
The sum of the descent differences of a permutations.
St000031Permutations ⟶ ℤ
The number of cycles in the cycle decomposition of a permutation.
St000033Permutations ⟶ ℤ
The number of permutations greater than or equal to the given permutation in (strong) Bruhat order.
St000034Permutations ⟶ ℤ
The maximum defect over any reduced expression for a permutation and any subexpression.
St000035Permutations ⟶ ℤ
The number of left outer peaks of a permutation.
St000036Permutations ⟶ ℤ
The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation.
St000037Permutations ⟶ ℤ
The sign of a permutation.
St000039Permutations ⟶ ℤ
The number of crossings of a permutation.
St000040Permutations ⟶ ℤ
The number of regions of the inversion arrangement of a permutation.
St000054Permutations ⟶ ℤ
The first entry of the permutation.
St000055Permutations ⟶ ℤ
The inversion sum of a permutation.
St000056Permutations ⟶ ℤ
The decomposition (or block) number of a permutation.
St000058Permutations ⟶ ℤ
The order of a permutation.
St000060Permutations ⟶ ℤ
The greater neighbor of the maximum.
St000062Permutations ⟶ ℤ
The length of the longest increasing subsequence of the permutation.
St000064Permutations ⟶ ℤ
The number of one-box pattern of a permutation.
St000078Permutations ⟶ ℤ
The number of alternating sign matrices whose left key is the permutation.
St000092Permutations ⟶ ℤ
The number of outer peaks of a permutation.
St000099Permutations ⟶ ℤ
The number of valleys of a permutation, including the boundary.
St000109Permutations ⟶ ℤ
The number of elements less than or equal to the given element in Bruhat order.
St000110Permutations ⟶ ℤ
The number of permutations less than or equal to a permutation in left weak order.
St000111Permutations ⟶ ℤ
The sum of the descent tops (or Genocchi descents) of a permutation.
St000119Permutations ⟶ ℤ
The number of occurrences of the pattern 321 in a permutation.
St000123Permutations ⟶ ℤ
The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map.
St000124Permutations ⟶ ℤ
The cardinality of the preimage of the Simion-Schmidt map.
St000133Permutations ⟶ ℤ
The "bounce" of a permutation.
St000141Permutations ⟶ ℤ
The maximum drop size of a permutation.
St000153Permutations ⟶ ℤ
The number of adjacent cycles of a permutation.
St000154Permutations ⟶ ℤ
The sum of the descent bottoms of a permutation.
St000155Permutations ⟶ ℤ
The number of exceedances (also excedences) of a permutation.
St000156Permutations ⟶ ℤ
The Denert index of a permutation.
St000162Permutations ⟶ ℤ
The number of nontrivial cycles in the cycle decomposition of a permutation.
St000209Permutations ⟶ ℤ
Maximum difference of elements in cycles.
St000210Permutations ⟶ ℤ
Minimum over maximum difference of elements in cycles.
St000213Permutations ⟶ ℤ
The number of weak exceedances (also weak excedences) of a permutation.
St000214Permutations ⟶ ℤ
The number of adjacencies of a permutation.
St000215Permutations ⟶ ℤ
The number of adjacencies of a permutation, zero appended.
St000216Permutations ⟶ ℤ
The absolute length of a permutation.
St000217Permutations ⟶ ℤ
The number of occurrences of the pattern 312 in a permutation.
St000218Permutations ⟶ ℤ
The number of occurrences of the pattern 213 in a permutation.
St000219Permutations ⟶ ℤ
The number of occurrences of the pattern 231 in a permutation.
St000220Permutations ⟶ ℤ
The number of occurrences of the pattern 132 in a permutation.
St000221Permutations ⟶ ℤ
The number of strong fixed points of a permutation.
St000222Permutations ⟶ ℤ
The number of alignments in the permutation.
St000223Permutations ⟶ ℤ
The number of nestings in the permutation.
St000224Permutations ⟶ ℤ
The sorting index of a permutation.
St000226Permutations ⟶ ℤ
The convexity of a permutation.
St000234Permutations ⟶ ℤ
The number of global ascents of a permutation.
St000235Permutations ⟶ ℤ
The number of indices that are not cyclical small weak excedances.
St000236Permutations ⟶ ℤ
The number of cyclical small weak excedances.
St000237Permutations ⟶ ℤ
The number of small exceedances.
St000238Permutations ⟶ ℤ
The number of indices that are not small weak excedances.
St000239Permutations ⟶ ℤ
The number of small weak excedances.
St000240Permutations ⟶ ℤ
The number of indices that are not small excedances.
St000241Permutations ⟶ ℤ
The number of cyclical small excedances.
St000242Permutations ⟶ ℤ
The number of indices that are not cyclical small weak excedances.
St000243Permutations ⟶ ℤ
The number of cyclic valleys and cyclic peaks of a permutation.
St000245Permutations ⟶ ℤ
The number of ascents of a permutation.
St000246Permutations ⟶ ℤ
The number of non-inversions of a permutation.
St000255Permutations ⟶ ℤ
The number of reduced Kogan faces with the permutation as type.
St000279Permutations ⟶ ℤ
The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations.
St000280Permutations ⟶ ℤ
The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations.
St000304Permutations ⟶ ℤ
The load of a permutation.
St000305Permutations ⟶ ℤ
The inverse major index of a permutation.
St000308Permutations ⟶ ℤ
The height of the tree associated to a permutation.
St000314Permutations ⟶ ℤ
The number of left-to-right-maxima of a permutation.
St000316Permutations ⟶ ℤ
The number of non-left-to-right-maxima of a permutation.
St000317Permutations ⟶ ℤ
The cycle descent number of a permutation.
St000324Permutations ⟶ ℤ
The shape of the tree associated to a permutation.
St000325Permutations ⟶ ℤ
The width of the tree associated to a permutation.
St000333Permutations ⟶ ℤ
The dez statistic, the number of descents of a permutation after replacing fixed points by zeros.
St000334Permutations ⟶ ℤ
The maz index, the major index of a permutation after replacing fixed points by zeros.
St000337Permutations ⟶ ℤ
The lec statistic, the sum of the inversion numbers of the hook factors of a permutation.
St000338Permutations ⟶ ℤ
The number of pixed points of a permutation.
St000339Permutations ⟶ ℤ
The maf index of a permutation.
St000341Permutations ⟶ ℤ
The non-inversion sum of a permutation.
St000342Permutations ⟶ ℤ
The cosine of a permutation.
St000352Permutations ⟶ ℤ
The Elizalde-Pak rank of a permutation.
St000353Permutations ⟶ ℤ
The number of inner valleys of a permutation.
St000354Permutations ⟶ ℤ
The number of recoils of a permutation.
St000355Permutations ⟶ ℤ
The number of occurrences of the pattern 21-3.
St000356Permutations ⟶ ℤ
The number of occurrences of the pattern 13-2.
St000357Permutations ⟶ ℤ
The number of occurrences of the pattern 12-3.
St000358Permutations ⟶ ℤ
The number of occurrences of the pattern 31-2.
St000359Permutations ⟶ ℤ
The number of occurrences of the pattern 23-1.
St000360Permutations ⟶ ℤ
The number of occurrences of the pattern 32-1.
St000365Permutations ⟶ ℤ
The number of double ascents of a permutation.
St000366Permutations ⟶ ℤ
The number of double descents of a permutation.
St000367Permutations ⟶ ℤ
The number of simsun double descents of a permutation.
St000371Permutations ⟶ ℤ
The number of mid points of decreasing subsequences of length 3 in a permutation.
St000372Permutations ⟶ ℤ
The number of mid points of increasing subsequences of length 3 in a permutation.
St000373Permutations ⟶ ℤ
The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$.
St000374Permutations ⟶ ℤ
The number of exclusive right-to-left minima of a permutation.
St000375Permutations ⟶ ℤ
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
St000401Permutations ⟶ ℤ
The size of the symmetry class of a permutation.
St000402Permutations ⟶ ℤ
Half the size of the symmetry class of a permutation.
St000404Permutations ⟶ ℤ
The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation.
St000405Permutations ⟶ ℤ
The number of occurrences of the pattern 1324 in a permutation.
St000406Permutations ⟶ ℤ
The number of occurrences of the pattern 3241 in a permutation.
St000407Permutations ⟶ ℤ
The number of occurrences of the pattern 2143 in a permutation.
St000408Permutations ⟶ ℤ
The number of occurrences of the pattern 4231 in a permutation.
St000423Permutations ⟶ ℤ
The number of occurrences of the pattern 123 or of the pattern 132 in a permutation.
St000424Permutations ⟶ ℤ
The number of occurrences of the pattern 132 or of the pattern 231 in a permutation.
St000425Permutations ⟶ ℤ
The number of occurrences of the pattern 132 or of the pattern 213 in a permutation.
St000426Permutations ⟶ ℤ
The number of occurrences of the pattern 132 or of the pattern 312 in a permutation.
St000427Permutations ⟶ ℤ
The number of occurrences of the pattern 123 or of the pattern 231 in a permutation.
St000428Permutations ⟶ ℤ
The number of occurrences of the pattern 123 or of the pattern 213 in a permutation.
St000429Permutations ⟶ ℤ
The number of occurrences of the pattern 123 or of the pattern 321 in a permutation.
St000430Permutations ⟶ ℤ
The number of occurrences of the pattern 123 or of the pattern 312 in a permutation.
St000431Permutations ⟶ ℤ
The number of occurrences of the pattern 213 or of the pattern 321 in a permutation.
St000432Permutations ⟶ ℤ
The number of occurrences of the pattern 231 or of the pattern 312 in a permutation.
St000433Permutations ⟶ ℤ
The number of occurrences of the pattern 132 or of the pattern 321 in a permutation.
St000434Permutations ⟶ ℤ
The number of occurrences of the pattern 213 or of the pattern 312 in a permutation.
St000435Permutations ⟶ ℤ
The number of occurrences of the pattern 213 or of the pattern 231 in a permutation.
St000436Permutations ⟶ ℤ
The number of occurrences of the pattern 231 or of the pattern 321 in a permutation.
St000437Permutations ⟶ ℤ
The number of occurrences of the pattern 312 or of the pattern 321 in a permutation.
St000440Permutations ⟶ ℤ
The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation.
St000441Permutations ⟶ ℤ
The number of successions of a permutation.
St000446Permutations ⟶ ℤ
The disorder of a permutation.
St000451Permutations ⟶ ℤ
The length of the longest pattern of the form k 1 2.
St000457Permutations ⟶ ℤ
The number of occurrences of one of the patterns 132, 213 or 321 in a permutation.
St000458Permutations ⟶ ℤ
The number of permutations obtained by switching adjacencies or successions.
St000461Permutations ⟶ ℤ
The rix statistic of a permutation.
St000462Permutations ⟶ ℤ
The major index minus the number of excedences of a permutation.
St000463Permutations ⟶ ℤ
The number of admissible inversions of a permutation.
St000470Permutations ⟶ ℤ
The number of runs in a permutation.
St000471Permutations ⟶ ℤ
The sum of the ascent tops of a permutation.
St000472Permutations ⟶ ℤ
The sum of the ascent bottoms of a permutation.
St000483Permutations ⟶ ℤ
The number of times a permutation switches from increasing to decreasing or decreasing to increasing.
St000484Permutations ⟶ ℤ
The sum of St000483 over all subsequences of length at least three.
St000485Permutations ⟶ ℤ
The length of the longest cycle of a permutation.
St000486Permutations ⟶ ℤ
The number of cycles of length at least 3 of a permutation.
St000487Permutations ⟶ ℤ
The length of the shortest cycle of a permutation.
St000488Permutations ⟶ ℤ
The number of cycles of a permutation of length at most 2.
St000489Permutations ⟶ ℤ
The number of cycles of a permutation of length at most 3.
St000494Permutations ⟶ ℤ
The number of inversions of distance at most 3 of a permutation.
St000495Permutations ⟶ ℤ
The number of inversions of distance at most 2 of a permutation.
St000500Permutations ⟶ ℤ
Eigenvalues of the random-to-random operator acting on the regular representation.
St000501Permutations ⟶ ℤ
The size of the first part in the decomposition of a permutation.
St000516Permutations ⟶ ℤ
The number of stretching pairs of a permutation.
St000520Permutations ⟶ ℤ
The number of patterns in a permutation.
St000530Permutations ⟶ ℤ
The number of permutations with the same descent word as the given permutation.
St000534Permutations ⟶ ℤ
The number of 2-rises of a permutation.
St000538Permutations ⟶ ℤ
The number of even inversions of a permutation.
St000539Permutations ⟶ ℤ
The number of odd inversions of a permutation.
St000541Permutations ⟶ ℤ
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
St000542Permutations ⟶ ℤ
The number of left-to-right-minima of a permutation.
St000545Permutations ⟶ ℤ
The number of parabolic double cosets with minimal element being the given permutation.
St000546Permutations ⟶ ℤ
The number of global descents of a permutation.
St000570Permutations ⟶ ℤ
The Edelman-Greene number of a permutation.
St000616Permutations ⟶ ℤ
The inversion index of a permutation.
St000619Permutations ⟶ ℤ
The number of cyclic descents of a permutation.
St000622Permutations ⟶ ℤ
The number of occurrences of the patterns 2143 or 4231 in a permutation.
St000623Permutations ⟶ ℤ
The number of occurrences of the pattern 52341 in a permutation.
St000624Permutations ⟶ ℤ
The normalized sum of the minimal distances to a greater element.
St000625Permutations ⟶ ℤ
The sum of the minimal distances to a greater element.
St000638Permutations ⟶ ℤ
The number of up-down runs of a permutation.
St000646Permutations ⟶ ℤ
The number of big ascents of a permutation.
St000647Permutations ⟶ ℤ
The number of big descents of a permutation.
St000648Permutations ⟶ ℤ
The number of 2-excedences of a permutation.
St000649Permutations ⟶ ℤ
The number of 3-excedences of a permutation.
St000650Permutations ⟶ ℤ
The number of 3-rises of a permutation.
St000651Permutations ⟶ ℤ
The maximal size of a rise in a permutation.
St000652Permutations ⟶ ℤ
The maximal difference between successive positions of a permutation.
St000653Permutations ⟶ ℤ
The last descent of a permutation.
St000654Permutations ⟶ ℤ
The first descent of a permutation.
St000662Permutations ⟶ ℤ
The staircase size of the code of a permutation.
St000663Permutations ⟶ ℤ
The number of right floats of a permutation.
St000664Permutations ⟶ ℤ
The number of right ropes of a permutation.
St000665Permutations ⟶ ℤ
The number of rafts of a permutation.
St000666Permutations ⟶ ℤ
The number of right tethers of a permutation.
St000669Permutations ⟶ ℤ
The number of permutations obtained by switching ascents or descents of size 2.
St000670Permutations ⟶ ℤ
The reversal length of a permutation.
St000672Permutations ⟶ ℤ
The number of minimal elements in Bruhat order not less than the permutation.
St000673Permutations ⟶ ℤ
The size of the support of a permutation.
St000677Permutations ⟶ ℤ
The standardized bi-alternating inversion number of a permutation.
St000690Permutations ⟶ ℤ
The size of the conjugacy class of a permutation.
St000692Permutations ⟶ ℤ
Babson and Steingrímsson's statistic of a permutation.
St000694Permutations ⟶ ℤ
The number of affine bounded permutations that project to a given permutation.
St000696Permutations ⟶ ℤ
The number of cycles in the breakpoint graph of a permutation.
St000702Permutations ⟶ ℤ
The number of weak deficiencies of a permutation.
St000703Permutations ⟶ ℤ
The number of deficiencies of a permutation.
St000709Permutations ⟶ ℤ
The number of occurrences of 14-2-3 or 14-3-2.
St000710Permutations ⟶ ℤ
The number of big deficiencies of a permutation.
St000711Permutations ⟶ ℤ
The number of big exceedences of a permutation.
St000724Permutations ⟶ ℤ
The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation.
St000725Permutations ⟶ ℤ
The smallest label of a leaf of the increasing binary tree associated to a permutation.
St000726Permutations ⟶ ℤ
The normalized sum of the leaf labels of the increasing binary tree associated to a permutation.
St000727Permutations ⟶ ℤ
The largest label of a leaf in the binary search tree associated with the permutation.
St000731Permutations ⟶ ℤ
The number of double exceedences of a permutation.
St000732Permutations ⟶ ℤ
The number of double deficiencies of a permutation.
St000740Permutations ⟶ ℤ
The last entry of a permutation.
St000742Permutations ⟶ ℤ
The number of big ascents of a permutation after prepending zero.
St000750Permutations ⟶ ℤ
The number of occurrences of the pattern 4213 in a permutation.
St000751Permutations ⟶ ℤ
The number of occurrences of either of the pattern 2143 or 2143 in a permutation.
St000756Permutations ⟶ ℤ
The sum of the positions of the left to right maxima of a permutation.
St000779Permutations ⟶ ℤ
The tier of a permutation.
St000794Permutations ⟶ ℤ
The mak of a permutation.
St000795Permutations ⟶ ℤ
The mad of a permutation.
St000796Permutations ⟶ ℤ
The stat' of a permutation.
St000797Permutations ⟶ ℤ
The stat`` of a permutation.
St000798Permutations ⟶ ℤ
The makl of a permutation.
St000799Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |213 in a permutation.
St000800Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |231 in a permutation.
St000801Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |312 in a permutation.
St000802Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |321 in a permutation.
St000803Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |132 in a permutation.
St000804Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |123 in a permutation.
St000809Permutations ⟶ ℤ
The reduced reflection length of the permutation.
St000824Permutations ⟶ ℤ
The sum of the number of descents and the number of recoils of a permutation.
St000825Permutations ⟶ ℤ
The sum of the major and the inverse major index of a permutation.
St000828Permutations ⟶ ℤ
The spearman's rho of a permutation and the identity permutation.
St000829Permutations ⟶ ℤ
The Ulam distance of a permutation to the identity permutation.
St000830Permutations ⟶ ℤ
The total displacement of a permutation.
St000831Permutations ⟶ ℤ
The number of indices that are either descents or recoils.
St000832Permutations ⟶ ℤ
The number of permutations obtained by reversing blocks of three consecutive numbers.
St000833Permutations ⟶ ℤ
The comajor index of a permutation.
St000834Permutations ⟶ ℤ
The number of right outer peaks of a permutation.
St000836Permutations ⟶ ℤ
The number of descents of distance 2 of a permutation.
St000837Permutations ⟶ ℤ
The number of ascents of distance 2 of a permutation.
St000842Permutations ⟶ ℤ
The breadth of a permutation.
St000844Permutations ⟶ ℤ
The size of the largest block in the direct sum decomposition of a permutation.
St000862Permutations ⟶ ℤ
The number of parts of the shifted shape of a permutation.
St000863Permutations ⟶ ℤ
The length of the first row of the shifted shape of a permutation.
St000864Permutations ⟶ ℤ
The number of circled entries of the shifted recording tableau of a permutation.
St000866Permutations ⟶ ℤ
The number of admissible inversions of a permutation in the sense of Shareshian-Wachs.
St000868Permutations ⟶ ℤ
The aid statistic in the sense of Shareshian-Wachs.
St000871Permutations ⟶ ℤ
The number of very big ascents of a permutation.
St000872Permutations ⟶ ℤ
The number of very big descents of a permutation.
St000873Permutations ⟶ ℤ
The aix statistic of a permutation.
St000879Permutations ⟶ ℤ
The number of long braid edges in the graph of braid moves of a permutation.
St000880Permutations ⟶ ℤ
The number of connected components of long braid edges in the graph of braid moves of a permutation.
St000881Permutations ⟶ ℤ
The number of short braid edges in the graph of braid moves of a permutation.
St000882Permutations ⟶ ℤ
The number of connected components of short braid edges in the graph of braid moves of a permutation.
St000883Permutations ⟶ ℤ
The number of longest increasing subsequences of a permutation.
St000884Permutations ⟶ ℤ
The number of isolated descents of a permutation.
St000886Permutations ⟶ ℤ
The number of permutations with the same antidiagonal sums.
St000887Permutations ⟶ ℤ
The maximal number of nonzero entries on a diagonal of a permutation matrix.
St000891Permutations ⟶ ℤ
The number of distinct diagonal sums of a permutation matrix.
St000923Permutations ⟶ ℤ
The minimal number with no two order isomorphic substrings of this length in a permutation.
St000956Permutations ⟶ ℤ
The maximal displacement of a permutation.
St000957Permutations ⟶ ℤ
The number of Bruhat lower covers of a permutation.
St000958Permutations ⟶ ℤ
The number of Bruhat factorizations of a permutation.
St000959Permutations ⟶ ℤ
The number of strong Bruhat factorizations of a permutation.
St000961Permutations ⟶ ℤ
The shifted major index of a permutation.
St000962Permutations ⟶ ℤ
The 3-shifted major index of a permutation.
St000963Permutations ⟶ ℤ
The 2-shifted major index of a permutation.
St000988Permutations ⟶ ℤ
The orbit size of a permutation under Foata's bijection.
St000989Permutations ⟶ ℤ
The number of final rises of a permutation.
St000990Permutations ⟶ ℤ
The first ascent of a permutation.
St000991Permutations ⟶ ℤ
The number of right-to-left minima of a permutation.
St000994Permutations ⟶ ℤ
The number of cycle peaks and the number of cycle valleys of a permutation.
St000996Permutations ⟶ ℤ
The number of exclusive left-to-right maxima of a permutation.
St001004Permutations ⟶ ℤ
The number of indices that are either left-to-right maxima or right-to-left minima.
St001005Permutations ⟶ ℤ
The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.
St001052Permutations ⟶ ℤ
The length of the exterior of a permutation.
St001059Permutations ⟶ ℤ
Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation.
St001061Permutations ⟶ ℤ
The number of indices that are both descents and recoils of a permutation.
St001074Permutations ⟶ ℤ
The number of inversions of the cyclic embedding of a permutation.
St001076Permutations ⟶ ℤ
The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12).
St001077Permutations ⟶ ℤ
The prefix exchange distance of a permutation.
St001078Permutations ⟶ ℤ
The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,.
St001079Permutations ⟶ ℤ
The minimal length of a factorization of a permutation using the permutations (12)(34).
St001080Permutations ⟶ ℤ
The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,.
St001081Permutations ⟶ ℤ
The number of minimal length factorizations of a permutation into star transpositions.
St001082Permutations ⟶ ℤ
The number of boxed occurrences of 123 in a permutation.
St001083Permutations ⟶ ℤ
The number of boxed occurrences of 132 in a permutation.
St001084Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |1-23 in a permutation.
St001085Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |21-3 in a permutation.
St001086Permutations ⟶ ℤ
The number of occurrences of the consecutive pattern 132 in a permutation.
St001087Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |12-3 in a permutation.
St001090Permutations ⟶ ℤ
The number of pop-stack-sorts needed to sort a permutation.
St001096Permutations ⟶ ℤ
The size of the overlap set of a permutation.
St001114Permutations ⟶ ℤ
The number of odd descents of a permutation.
St001115Permutations ⟶ ℤ
The number of even descents of a permutation.
St001130Permutations ⟶ ℤ
The number of two successive successions in a permutation.
St001160Permutations ⟶ ℤ
The number of proper blocks (or intervals) of a permutations.
St001162Permutations ⟶ ℤ
The minimum jump of a permutation.
St001168Permutations ⟶ ℤ
The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
St001171Permutations ⟶ ℤ
The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$.
St001174Permutations ⟶ ℤ
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
St001207Permutations ⟶ ℤ
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
St001208Permutations ⟶ ℤ
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
St001220Permutations ⟶ ℤ
The width of a permutation.
St001245Permutations ⟶ ℤ
The cyclic maximal difference between two consecutive entries of a permutation.
St001246Permutations ⟶ ℤ
The maximal difference between two consecutive entries of a permutation.
St001269Permutations ⟶ ℤ
The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.
St001285Permutations ⟶ ℤ
The number of primes in the column sums of the two line notation of a permutation.
St001287Permutations ⟶ ℤ
The number of primes obtained by multiplying preimage and image of a permutation and subtracting one.
St001288Permutations ⟶ ℤ
The number of primes obtained by multiplying preimage and image of a permutation and adding one.
St001293Permutations ⟶ ℤ
The sum of all $1/(i+\pi(i))$ for a permutation $\pi$ times the lcm of all possible values among permutations of the same length.
St001298Permutations ⟶ ℤ
The number of repeated entries in the Lehmer code of a permutation.
St001332Permutations ⟶ ℤ
The number of steps on the non-negative side of the walk associated with the permutation.
St001344Permutations ⟶ ℤ
The neighbouring number of a permutation.
St001346Permutations ⟶ ℤ
The number of parking functions that give the same permutation.
St001359Permutations ⟶ ℤ
The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles.
St001375Permutations ⟶ ℤ
The pancake length of a permutation.
St001377Permutations ⟶ ℤ
The major index minus the number of inversions of a permutation.
St001379Permutations ⟶ ℤ
The number of inversions plus the major index of a permutation.
St001381Permutations ⟶ ℤ
The fertility of a permutation.
St001388Permutations ⟶ ℤ
The number of non-attacking neighbors of a permutation.
St001390Permutations ⟶ ℤ
The number of bumps occurring when Schensted-inserting the letter 1 of a permutation.
St001394Permutations ⟶ ℤ
The genus of a permutation.
St001402Permutations ⟶ ℤ
The number of separators in a permutation.
St001403Permutations ⟶ ℤ
The number of vertical separators in a permutation.
St001405Permutations ⟶ ℤ
The number of bonds in a permutation.
St001411Permutations ⟶ ℤ
The number of patterns 321 or 3412 in a permutation.
St001412Permutations ⟶ ℤ
Number of minimal entries in the Bruhat order matrix of a permutation.
St001439Permutations ⟶ ℤ
The number of even weak deficiencies and of odd weak exceedences.
St001461Permutations ⟶ ℤ
The number of topologically connected components of the chord diagram of a permutation.
St001464Permutations ⟶ ℤ
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
St001465Permutations ⟶ ℤ
The number of adjacent transpositions in the cycle decomposition of a permutation.
St001466Permutations ⟶ ℤ
The number of transpositions swapping cyclically adjacent numbers in a permutation.
St001468Permutations ⟶ ℤ
The smallest fixpoint of a permutation.
St001469Permutations ⟶ ℤ
The holeyness of a permutation.
St001470Permutations ⟶ ℤ
The cyclic holeyness of a permutation.
St001482Permutations ⟶ ℤ
The product of the prefix sums of a permutation.
St001489Permutations ⟶ ℤ
The maximum of the number of descents and the number of inverse descents.
St001497Permutations ⟶ ℤ
The position of the largest weak excedence of a permutation.
St001511Permutations ⟶ ℤ
The minimal number of transpositions needed to sort a permutation in either direction.
St001513Permutations ⟶ ℤ
The number of nested exceedences of a permutation.
St001516Permutations ⟶ ℤ
The number of cyclic bonds of a permutation.
St001517Permutations ⟶ ℤ
The length of a longest pair of twins in a permutation.
St001519Permutations ⟶ ℤ
The pinnacle sum of a permutation.
St001520Permutations ⟶ ℤ
The number of strict 3-descents.
St001528Permutations ⟶ ℤ
The number of permutations such that the product with the permutation has the same number of fixed points.
St001535Permutations ⟶ ℤ
The number of cyclic alignments of a permutation.
St001536Permutations ⟶ ℤ
The number of cyclic misalignments of a permutation.
St001537Permutations ⟶ ℤ
The number of cyclic crossings of a permutation.
St001549Permutations ⟶ ℤ
The number of restricted non-inversions between exceedances.
St001550Permutations ⟶ ℤ
The number of inversions between exceedances where the greater exceedance is linked.
St001551Permutations ⟶ ℤ
The number of restricted non-inversions between exceedances where the rightmost exceedance is linked.
St001552Permutations ⟶ ℤ
The number of inversions between excedances and fixed points of a permutation.
St001556Permutations ⟶ ℤ
The number of inversions of the third entry of a permutation.
St001557Permutations ⟶ ℤ
The number of inversions of the second entry of a permutation.
St001558Permutations ⟶ ℤ
The number of transpositions that are smaller or equal to a permutation in Bruhat order.
St001559Permutations ⟶ ℤ
The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions.
St001560Permutations ⟶ ℤ
The product of the cardinalities of the lower order ideal and upper order ideal generated by a permutation in weak order.
St001565Permutations ⟶ ℤ
The number of arithmetic progressions of length 2 in a permutation.
St001566Permutations ⟶ ℤ
The length of the longest arithmetic progression in a permutation.
St001569Permutations ⟶ ℤ
The maximal modular displacement of a permutation.
St001579Permutations ⟶ ℤ
The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation.
St001582Permutations ⟶ ℤ
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
St001583Permutations ⟶ ℤ
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
St001639Permutations ⟶ ℤ
The number of alternating subsets such that applying the permutation does not yield an alternating subset.
St001640Permutations ⟶ ℤ
The number of ascent tops in the permutation such that all smaller elements appear before.
St001652Permutations ⟶ ℤ
The length of a longest interval of consecutive numbers.
St001661Permutations ⟶ ℤ
Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation.
St001662Permutations ⟶ ℤ
The length of the longest factor of consecutive numbers in a permutation.
St001663Permutations ⟶ ℤ
The number of occurrences of the Hertzsprung pattern 132 in a permutation.
St001665Permutations ⟶ ℤ
The number of pure excedances of a permutation.
St001667Permutations ⟶ ℤ
The maximal size of a pair of weak twins for a permutation.
St001671Permutations ⟶ ℤ
Haglund's hag of a permutation.
St001682Permutations ⟶ ℤ
The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation.
St001683Permutations ⟶ ℤ
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
St001684Permutations ⟶ ℤ
The reduced word complexity of a permutation.
St001685Permutations ⟶ ℤ
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
St001687Permutations ⟶ ℤ
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
St001705Permutations ⟶ ℤ
The number of occurrences of the pattern 2413 in a permutation.
St001715Permutations ⟶ ℤ
The number of non-records in a permutation.
St001726Permutations ⟶ ℤ
The number of visible inversions of a permutation.
St001727Permutations ⟶ ℤ
The number of invisible inversions of a permutation.
St001728Permutations ⟶ ℤ
The number of invisible descents of a permutation.
St001729Permutations ⟶ ℤ
The number of visible descents of a permutation.
St001731Permutations ⟶ ℤ
The factorization defect of a permutation.
St001735Permutations ⟶ ℤ
The number of permutations with the same set of runs.
St001737Permutations ⟶ ℤ
The number of descents of type 2 in a permutation.
St001741Permutations ⟶ ℤ
The largest integer such that all patterns of this size are contained in the permutation.
St001744Permutations ⟶ ℤ
The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation.
St001745Permutations ⟶ ℤ
The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation.
St001759Permutations ⟶ ℤ
The Rajchgot index of a permutation.
St001760Permutations ⟶ ℤ
The number of prefix or suffix reversals needed to sort a permutation.
St001761Permutations ⟶ ℤ
The maximal multiplicity of a letter in a reduced word of a permutation.
St001766Permutations ⟶ ℤ
The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation.
St001778Permutations ⟶ ℤ
The largest greatest common divisor of an element and its image in a permutation.
St001801Permutations ⟶ ℤ
Half the number of preimage-image pairs of different parity in a permutation.
St001806Permutations ⟶ ℤ
The upper middle entry of a permutation.
St001807Permutations ⟶ ℤ
The lower middle entry of a permutation.
St001810Permutations ⟶ ℤ
The number of fixed points of a permutation smaller than its largest moved point.
St001811Permutations ⟶ ℤ
The Castelnuovo-Mumford regularity of a permutation.
St001847Permutations ⟶ ℤ
The number of occurrences of the pattern 1432 in a permutation.
St001850Permutations ⟶ ℤ
The number of Hecke atoms of a permutation.
St001856Permutations ⟶ ℤ
The number of edges in the reduced word graph of a permutation.
St001859Permutations ⟶ ℤ
The number of factors of the Stanley symmetric function associated with a permutation.
St001874Permutations ⟶ ℤ
Lusztig's a-function for the symmetric group.
St001906Permutations ⟶ ℤ
Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation.
St001911Permutations ⟶ ℤ
A descent variant minus the number of inversions.
St001928Permutations ⟶ ℤ
The number of non-overlapping descents in a permutation.
St001941Permutations ⟶ ℤ
The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5.
St001948Permutations ⟶ ℤ
The number of augmented double ascents of a permutation.
St001958Permutations ⟶ ℤ
The degree of the polynomial interpolating the values of a permutation.