Triangles Row Sum Factorial

Triangles summing to the factorial numbers AND having the factorial numbers as diagonal

A008275 Triangle read by rows of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.

A008276 Triangle of Stirling numbers of first kind, s(n,n-k+1), n >= 1, 1<=k<=n. Also triangle T(n,k) giving coefficients in expansion of n!*C(x,n)/x in powers of x.

A048994 Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0<=k<=n.

A054115 Triangular array generated by its row sums: T(n,0)=1 for n >= 1, T(n,1)=r(n-1), T(n,k)=T(n,k-1)+r(n-k) for k=2,3,...,n, n >= 2, r(h)=sum of the numbers in row h of T.

A054654 Triangle read by rows: matrix product of the binomial coefficients with the Stirling numbers of the first kind.

A056151 Distribution of maximum inversion table entry.

A094638 Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind (1<=k<=n; in other words, the unsigned Stirling numbers of the first kind in reverse order).

A100822 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n with k cells in the first column. (A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column).

A109878 Triangle read by rows: see below.

A116854 Triangle T(n,k) = A116853(n,k) - A116853(n,k-1) read by rows.

A126074 Triangle read by rows: T(n,k) is the number of permutations of n elements that have the longest cycle length k.

A130534 Triangle T(n,k), 0<=k<=n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1,k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.

A132393 Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows.

A134433 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which the last entry of the first increasing run is equal to k (1<=k<=n).

A134436 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n with k cells in the second row (0<=k<=n-1; a deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column).

A134830 Triangle of rank k of permutations of {1,2,...,n}.

A136125 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which the size of the last cycle is k (the cycles are ordered by increasing smallest elements; 1 <= k <=n).

A136572 Triangle read by rows: row n consists of n zeros followed by n!.

A138771 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} whose 2nd cycle has k entries; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements (n>=1; 0<=k<=n-1). For example, 1432=(1)(24)(3) has 2 entries in the 2nd cycle; 3421=(1324) has 0 entries in the 2nd cycle.

A141476 Triangle T(n,k) = A000142(n-k)*A003319(k+1) read by rows.

A144107 Eigentriangle, row sums = n!

A144108 Eigentriangle based on A052186 (permutations without strong fixed points), row sums = n!

A145877 Triangle read by rows: T(n,k) is the number of permutations of [n] for which the shortest cycle length is k (1<=k<=n).

A145888 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which k is the largest entry in the cycle containing 1 (1<=k<=n).

A202992 Triangle T(n,k), read by rows, given by (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...) DELTA (0, 0, 1, 1, 2, 2, 3, 3, 4, 4, ...) where DELTA is the operator defined in A084938.

Triangles summing to the factorial numbers

A001100 Triangle read by rows: T(n,k) = number of permutations of length n with exactly k rising or falling successions, for n >= 1, 0 <= k <= n-1.

A008275 Triangle read by rows of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.

A008276 Triangle of Stirling numbers of first kind, s(n,n-k+1), n >= 1, 1<=k<=n. Also triangle T(n,k) giving coefficients in expansion of n!*C(x,n)/x in powers of x.

A008290 Triangle T(n,k) of rencontres numbers (number of permutations of n elements with k fixed points).

A008292 Triangle of Eulerian numbers T(n,k) (n>=1, 1 <= k <= n) read by rows.

A008304 Triangle read by rows: T(n,k) (n>=1; 1<=k<=n) is the number of permutations of [n] in which the longest increasing run has length k.

A010026 Triangle read by rows: number of permutations of 1..n by length of longest run.

A010027 Triangle read by rows: T(n,k) is the number of permutations of [n] having k consecutive ascending pairs (0<=k<=n-1).

A028305 Triangle of numbers of permutations eliminating just k cards out of n in game of Mousetrap.

A047874 Triangle of numbers T(n,k) = number of permutations of (1,2,...,n) with longest increasing subsequence of length k (1<=k<=n).

A047918 Triangular array read by rows: a(n,k) = Sum_{d|k} mu(d)*U(n,k/d) if k|n else 0, where U(n,k)=A047916(n,k) (1<=k<=n).

A047919 Triangular array read by rows: a(n,k) = Sum_{d|k} mu(d)*U(n,k/d)/n if k|n else 0, where U(n,k)=A047916(n,k) (1<=k<=n).

A047921 Triangle of numbers a(n,k) = number of permutations on n letters containing k 3-sequences (n >= 2, 0<=k<=n-2).

A048994 Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0<=k<=n.

A054115 Triangular array generated by its row sums: T(n,0)=1 for n >= 1, T(n,1)=r(n-1), T(n,k)=T(n,k-1)+r(n-k) for k=2,3,...,n, n >= 2, r(h)=sum of the numbers in row h of T.

A054654 Triangle read by rows: matrix product of the binomial coefficients with the Stirling numbers of the first kind.

A056151 Distribution of maximum inversion table entry.

A058057 Triangle giving coefficients of ménage hit polynomials.

A058087 Triangle giving coefficients of ménage hit polynomials.

A059418 Triangle T(n,k) arising from enumeration of permutations with ordered orbits, read by rows (1<=k<=n).

A059427 Triangle read by rows: T(n,k) is the number of permutations of [n] with k alternating runs (n>=2, k>=1). The permutation 732569148 has 4 alternating runs: 732, 2569, 91 and 148.

A059438 Triangle T(n,k) (1<=k<=n) read by rows: T(n,k) = number of permutations of [1..n] with k components.

A060338 Triangle T(n,k) of coefficients of Meixner polynomials of degree n, k=0..n.

A060523 Triangle T(n,k) = number of degree-n permutations with k even cycles, k=0..n.

A060524 Triangle read by rows: T(n,k) = number of degree-n permutations with k odd cycles, k=0..n, n >= 0.

A062867 Triangle read by rows: entries give numbers of permutations of [1..n] by absolute barycenter.

A064315 Triangle of number of permutations by length of shortest ascending run.

A064482 Triangle read by rows: T(n,k) (n >= 2, 1<=k<=n-1) is the number of permutations p of 1,...,n with max(|p(i)-p(i-1)|,i=2..n) = k.

A064484 Triangle T(n,k), n >= 2, n+1 <= k <= 2*n-1, number of permutations p of 1,...,n, with max(p(i)+p(i-1), i=2..n) = k.

A071818 Triangle of T(n,k) where T(n,k)/(n-1)! is probability of player k out of n players winning a game of "Elimination": rules are that player 1 chooses one of the others at random to be eliminated, then player 2 (or 3 if player 2 has been eliminated) chooses somebody else at random from the survivors to be eliminated next, then the next surviving player chooses and so on round the circle until all but one have been eliminated.

A079642 Matrix product of unsigned Stirling1-triangle |A008275(n,k)| and Stirling1-triangle A008275(n,k).

A080018 Triangle of coefficients of polynomials P(n; x) = Permanent(M), where M=[m(i,j)] is n X n matrix defined by m(i,j)=x if -1<=i-j<=1 else m(i,j)=1.

A080061 Triangle of coefficients of polynomials P(n; x) = Permanent(M), where M=[m(i,j)] is n X n matrix defined by m(i,j)=x if 0<=i-j<=2 else m(i,j)=1.

A085771 Triangle A059438(n,k), 0<=k<=n, with an extra column of zeros.

A092580 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which exactly the first k terms satisfy the up-down property, i.e. p(1)< p(2), p(2)>p(3), p(3)<p(4), ...

A092582 Triangle read by rows: T(n,k) is the number of permutations p of [n] having length of first run equal to k.

A092583 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding the 123-pattern is equal to k.

A092594 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding both the 132- and the 231-pattern is equal to k.

A092741 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding both the 132- and the 321-pattern is equal to k.

A094067 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding the 123-, the 132- and the 321-pattern is equal to k.

A094112 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding the 123-, the 132- and the 231-pattern is equal to k.

A094314 Triangle read by rows: T(n,k) = number of ways of seating n couples around a circular table so that exactly k married couples are adjacent (0 <= k <= n).

A094315 Triangle read by rows giving number of circular permutations of n letters such that all letters are displaced by no more than k places from their original position.

A094638 Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind (1<=k<=n; in other words, the unsigned Stirling numbers of the first kind in reverse order).

A094785 Triangle read by rows: T(n,k) is the number of permutations p of [n] such that the length of the longest 2-stack sortable initial segment of p is equal to k.

A097591 Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of odd length.

A097898 Triangle read by rows: T(n,k) is the number of permutations of [n] with k runs of length 1. For example, 457/3/26/1 has two runs of length 1: 3 and 1.

A098825 Triangle read by rows: T(n,k) = number of partial derangements, that is, the number of permutations of n distinct, ordered items in which exactly k of the items are in their natural ordered positions, for n >= 0, k = n, n-1, ..., 1, 0.

A100822 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n with k cells in the first column. (A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column).

A109878 Triangle read by rows: see below.

A115755 Number of permutations of n elements whose unsigned reversal distance is k.

A116854 Triangle T(n,k) = A116853(n,k) - A116853(n,k-1) read by rows.

A120434 Triangle read by rows: counts permutations by number of big descents.

A121554 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 1-cell columns (0<=k<=n). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

A121581 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having k cells in the second column (n>=1, k>=0). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

A121585 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 1-cell columns starting at level 0 (0<=k<=n). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

A121634 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 2-cell columns starting at level 0 (n>=1; 0<=k<=n-1). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

A121637 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 2-cell columns (n>=1; 0<=k<=n-1). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

A121692 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and vertical height (i.i. number of rows) k (1<=k<=n). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

A121697 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns ending at an odd level (0<=k<=n). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

A121698 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns ending at an even level (1<=k<=n). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

A121745 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns of odd length (0<=k<=n). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

A121748 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns of even length (0<=k<=n). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

A122890 Triangle, read by rows, where the g.f. of row n divided by (1-x)^n yields the g.f. of column n in the triangle A122888, for n>=1.

A123125 Triangle of Eulerian numbers T(n,k), 0<=k<=n, read by rows.

A123513 Triangle read by rows: T(n,k) is the number of permutations of [n] having k small descents (n>=1; 0<=k<=n-1). A small descent in a permutation (x_1,x_2,...,x_n) is a position i such that x_i - x_(i+1) =1.

A125182 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} such that the set {p(i)-i, i=1,2,...,n} has exactly k elements (1<=k<=n).

A125183 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} such that the set {|p(i)-i|, i=1,2,...,n} has exactly k elements (1<=k<=n).

A125553 Triangle read by rows: T(n,k) = S1(n,k)*2^k, where S1(n,k) is an unsigned Stirling number of the first kind (cf. A008275) (n >= 1, 1 <= k <= n).

A126065 Triangle of numbers read by rows: T(n,k) = number of permutations sigma of (1,2,...,n) with n - {length of longest increasing subsequence in sigma} = k (0<=k<=n-1).

A126074 Triangle read by rows: T(n,k) is the number of permutations of n elements that have the longest cycle length k.

A126440 Triangular array read by rows: related to A053445 and A060351 with row sums A000142 (which counts permutations of n objects).

A130152 Triangle read by rows: T(n,k)=number of permutations p of [n] such that max(|p(i)-i|)=k (n>=1, 0<=k<=n-1).

A130477 Triangle generated from finite differences of A130461.

A130534 Triangle T(n,k), 0<=k<=n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1,k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.

A132005 Triangle, read by rows, where T(n,k) = n*T(n-1,k-1) + T(n-1,k-2) for n>0 and k>1, with T(n,0) = T(n-1,n-1) and T(n,1) = n*T(n-1,0) for n>0 and T(0,0) = 1.

A132393 Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows.

A132795 Triangle of Gely numbers, read by rows.

A134433 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which the last entry of the first increasing run is equal to k (1<=k<=n).

A134436 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n with k cells in the second row (0<=k<=n-1; a deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column).

A134830 Triangle of rank k of permutations of {1,2,...,n}.

A134832 Triangle of succession numbers for circular permutations.

A136125 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which the size of the last cycle is k (the cycles are ordered by increasing smallest elements; 1 <= k <=n).

A136572 Triangle read by rows: row n consists of n zeros followed by n!.

A136715 Triangle T(n,k), 1 <= k <= n, read by rows: T(n,k) is the number of permutations of the set {2,4,6,...,2n} with k excedances. Equivalently, T(n,k) is the number of permutations in the symmetric group S_n having k multiplicative 2-excedances.

A136716 Triangle T(n,k), 0 <= k < n, read by rows: T(n,k) is the number of permutations of the set O_n = {1,3,5,...,2n-1} with k excedances.

A136717 Triangle T(n,k), 1 <= k <= n, read by rows: T(n,k) is the number of permutations in the symmetric group S_n having k multiplicative 3-excedances. Equivalently, the number of permutations of the set {3,6,9,...,3n} with k excedances.

A137312 A triangular sequence from a coefficients of generalized factorial polynomial recursion from Roman:a=1/2; p(x, n) = (x/a - (n - 1))*p(x, n - 1).

A137320 A triangular sequence from a coefficients of a raising factorial polynomial sequence recursion: p(x, n) = (m*x + n - 1)*p(x, n - 1).

A138770 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} such that there are exactly k entries between the entries 1 and 2 (n>=2, 0<=k<=n-2).

A138771 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} whose 2nd cycle has k entries; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements (n>=1; 0<=k<=n-1). For example, 1432=(1)(24)(3) has 2 entries in the 2nd cycle; 3421=(1324) has 0 entries in the 2nd cycle.

A140709 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n in which the maximal number of initial consecutive columns ending at the same level is k (1<=k<=n). (A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column).

A140711 Irregular triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k white corners.

A141476 Triangle T(n,k) = A000142(n-k)*A003319(k+1) read by rows.

A144107 Eigentriangle, row sums = n!

A144108 Eigentriangle based on A052186 (permutations without strong fixed points), row sums = n!

A145876 Triangle read by rows: T(n,k) is the number of permutations of [n] having k-1 alternating descents (1<=k<=n). The index i is an alternating descent of a permutation p if either i is odd and p(i)>p(i+1), or i is even and p(i)<p(i+1).

A145877 Triangle read by rows: T(n,k) is the number of permutations of [n] for which the shortest cycle length is k (1<=k<=n).

A145878 Triangle read by rows: T(n,k) is the number of permutations of [n] having k strong fixed points (0<=k<=n). A permutation p of {1,2,...,n} is said to have j as a strong fixed point (splitter) if p(k)<j for k<j and p(k)>j for k>j.

A145888 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which k is the largest entry in the cycle containing 1 (1<=k<=n).

A145893 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} such that j and p(j) are of opposite parities for k values of j (0<=k<=n).

A145894 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} such that j and p(j) are of the same parity for k values of j (0<=k<=n).

A147679 Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= n-1) is the number of permutations of [0..(n-1)] of spread k.

A152660 Triangle read by rows: T(n,k) is the number of permutations of [n] for which k is the maximal number of initial entries whose parities alternate (1<=k<=n).

A152874 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} with k parity changes (n>=2; 1<=k <=n-1; the permutation 372185946 has 5 parity changes: 37-2-1-8-59-46.

A152937 A vector recursion designed around a factorial row sum : v(n)=if[odd,{1.n,n^2,...,n!-Sum[2^m,{m,0,n/2-1}],n!-Sum2^m,{m,0,n/2-1}],...n^2.n,1}],if[ even{1.n,n^2,...,n!-2Sum[2^m,{m,0,n/2-1}],...n^2.n,1}].

A152938 A vector recursion designed around a factorial row sum : v(n)=if[odd,{1.n,n^2,...,(n+1)!/2-Sum[2^m,{m,0,n/2-1}],(n+1)!/2-Sum2^m,{m,0,n/2-1}],...n^2.n,1}],if[ even{1.n,n^2,...,(n+1)!-2Sum[2^m,{m,0,n/2-1}],...n^2.n,1}].

A152970 A vector sequence with set row sum function: row(n)=(n+1)! and linear build up and decline function: f(n,m)=Floor[(m/n)*row(n)].

A153592 Triangle read by rows: T(n,k) = T(n-1,k-1) +T(n-1,k) +n*(n-1)*T(n-2,k-1) for n>4 and 1<=k<=n.

A154986 Polynomial recursion: p(x, n) = (x + 1)*p(x, n - 1) + (n^2 - n)*x*p(x, n - 2).

A155755 Triangle t(n,m)= A143491(n+2,m+2)+A143491(n+2,n-m+2) read by rows.

A156368 A ménage triangle.

A156996 A triangle sequence from polynomial coefficients: p(x,n)=If[n == 0, 1, Sum[Binomial[2*n - m, m]*(n - m)!*(2*n/(2*n - m))*(x - 1)^m, {m, 0, n}]].

A158830 Triangle, read by rows n>=1, where row n is the n-th differences of column n of array A158825, where the g.f. of row n of A158825 is the n-th iteration of x*Catalan(x).

A162976 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k double descents and initial descents (n>=1; 0<=k<=n-1) [we say that i is a doubledescent of a permutation p if p(i)>p(i+1)>p(i+2); we say that a permutation p has an initial descent if p(1)>p(2)].

A164645 Triangle read by rows: a(n,k) is the number of permutations of n elements with prefix transposition distance equal to k.

A164652 Triangle read by rows: Hultman numbers: a(n,k) is the number of permutations of n elements whose cycle graph (as defined by Bafna and Pevzner) contains k cycles.

A167565 A triangle related to the a(n) formulas for the rows of the ED2 array A167560.

A173018 Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.

A177262 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} starting with exactly k consecutive integers (1<=k<=n).

A177263 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k as the last entry in the first block (1<=k<=n). A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 45123867 has 4 blocks: 45, 123, 8, and 67.

A177264 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k as the first entry in the last block (1<=k<=n). A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 45123867 has 4 blocks: 45, 123, 8, and 67.

A177267 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having genus k (see first comment for definition of genus).

A179454 Permutation trees of power n and height k.

A180013 Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have exactly k cycles; n>=1, 1<=k<=n.

A180188 Triangle read by rows: T(n,k) is the number of permutations of [n] with k circular successions (0<=k<=n-1). A circular succession in a permutation p of [n] is either a pair p(i), p(i+1), where p(i+1)=p(i)+1 or the pair p(n), p(1) if p(1)=p(n)+1.

A180190 Triangle read by rows: T(n,k) is the number of permutations p of [n] for which k is the smallest among the positive differences p(i+1) - p(i); k=0 for the reversal of the identity permutation (0<=k<=n-1).

A180193 Triangle read by rows: T(n,k) is the number of permutations of [n] having k blocks of odd length (0<=k<=n).

A180196 Triangle read by rows: T(n,k) is the number of permutations of [n] that have k isolated entries (0<=k<=n).

A180887 Array read by antidiagonals: T(n,k)=number of permutations p() of 1..n+k with centered difference p(i+1)-p(i-1) < 0 exactly k-1 times

A182822 Exponential Riordan array, defining orthogonal polynomials related to permutations without double falls.

A184180 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} whose shortest block is of length k (1<=k<=n). A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 4512367 has 3 blocks: 45, 123, and 67. Its shortest block has length 2.

A184182 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} whose longest block is of length k (1<=k<=n). A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 5412367 has 4 blocks: 5, 4, 123, and 67. Its longest block has length 3.

A184184 Triangle read by rows: T(n,k) is the number of permutations of [n] having k adjacent cycles (0 <= k <= n). An adjacent cycle is a cycle of the form (i, i+1,i+2,...) (including 1-element cycles).

A186358 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k up-down cycles (0<=k<=n). A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1)<b(2)>b(3)<... .

A186370 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k up-down runs (1<=k<=n). The up-down runs of a permutation p are the alternating runs of the permutation p endowed with a 0 in the front. For example, 75814632 has 6 up-down runs: 07, 75, 58, 81, 146, and 632.

A186754 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing cycles (0<=k<=n). A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1) < b(2) < b(3) < ... .

A186759 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k cycles that are either nonincreasing or of length 1 (0<=k<=n). A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1) < b(2) < b(3) < ... .

A186761 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing odd cycles (0<=k<=n). A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)<b(2)<b(3)<... . A cycle is said to be odd if it has an odd number of entries. For example, the permutation (152)(347)(6)(8) has 3 increasing odd cycles.

A187247 Triangle read by rows: T(n,k) is the number of permutations of [n] having k cycles with at most 2 alternating runs (it is assumed that the smallest element of the cycle is in the first position), 0<=k<=n.

A191716 a(n,k) equals (1/n!) multiplied by the count of permutations with cycle length k in all products u v u^-1 v^-1 over all permutations u and v of length n.

A191718 a(n,k) is the count of permutations with cycle length k in the products w*w over all permutations w of length n.

A195581 Number T(n,k) of permutations of {1,2,...,n} that result in a binary search tree of height k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A199335 Triangle T(n,k), read by rows, given by (0,1,0,2,0,3,0,4,0,5,0,6,0,7,0,8,0,9,...) DELTA (2,0,3,0,4,0,5,0,6,0,7,0,8,0,9,...), where DELTA is the operator defined in A084938.

A200545 Triangle T(n,k), read by rows, given by (1,0,2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) DELTA (0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938.

A202992 Triangle T(n,k), read by rows, given by (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...) DELTA (0, 0, 1, 1, 2, 2, 3, 3, 4, 4, ...) where DELTA is the operator defined in A084938.

A208956 Triangular array read by rows. T(n,k) is the number of n-permutations that have at least k fixed points with n >= 1 and 1 <= k <= n.

A211318 Triangle read by rows: number of permutations of 1..n by length l of longest run (n >= 1, 1 <= l <= n).

A213166 Triangle, read by rows, of permutations of length n with k white global corners.

A216718 Triangle read by rows: number of circular permutations of [1..n] with k progressions of rise 1, distance 1 and length 3 (n >= 3, k >= 0).

A224652 Triangle read by rows: T(n,k) is the number of permutations of n elements with k the (smallest) header (first element) of the longest descending subsequence.

A233440 Triangle read by rows: T(n, k) = n*binomial(n, k)*A000757(k), 0 <= k <= n.

A235943 Number a(n,k) of positions (cyclic permutations) of circular permutations of [n] with exactly k (unspecified) increasing or decreasing modular runs (3-sequences), with clockwise and counterclockwise traversals counted as distinct; triangle a(n,k) read by rows, 0<=k<=n.

A244312 Triangle read by rows: T(n,k) is the number of single loop solutions formed by n proper arches (connecting an odd and even vertice from 1 to 2n) above the x axis, k arches above the x axis connecting an odd vertice to a higher even vertice and a rainbow of n arches below the x axis.

A245693 Number T(n,k) of permutations on [n] that are self-inverse on [k] but not on [k+1]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A259708 Triangle T(n,k) (0 <= k <= n) giving coefficients of certain polynomials related to Fibonacci numbers.

A261685 Triangle read by rows: T(n,m) = number of beta-labeled graphs of size n and order m (n>=1, 2<=m<=n+1).

A262494 Triangle read by rows: T(n,k) (n>=1, 0<=k<n) is the number of permutations of n things that require k stack-sorts.

A263484 Triangle read by rows: T(n,k) (n>=1, 0<=k<n) is the number of permutations of n with n! - k permutations in its connectivity set.

A263757 Triangle read by rows: T(n,k) (n>=1, 0<=k<n) is the number of permutations of n with maximal difference k between elements in the same cycle.

A264027 Triangle read by rows: T(n, k) = Sum_{t=k..n-2} (-1)^(t-k)*(n-t)!*binomial(t,k)*binomial(n-2,t).

A264028 Triangle read by rows: T(n, k) = Sum_{t=k..n-3} (-1)^(t-k)*(n-t)!*binomial(t,k)*binomial(n-3,t).