There are

(and possibly some waiting for verification)

**397 statistics**on**Permutations**in the database:(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 $i$ such that $\pi_i \neq i+1$ considered cyclically.

St000236Permutations ⟶ ℤ

The number of cyclical small weak excedances.

St000237Permutations ⟶ ℤ

The number of small exceedances.

St000238Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i \notin \{i,i+1\}$.

St000239Permutations ⟶ ℤ

The number of small weak excedances.

St000240Permutations ⟶ ℤ

The number of indices $i$ for which $\pi_i \neq i+1$.

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 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.