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# Triangles Row Sum Almost Factorial

## Classes of permutation

### Derangements and Rencontre numbers

Row sums : A000240 Rencontre numbers

A219836 Triangular array counting derangements by number of descents.

Row sums : A000166 Derangements (permutations without fixed points)

A046739 Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.

A061018 Triangle: a(n,m) = number of permutations of (1,2,...,n) with one or more fixed points in the m first positions.

A079510 Triangle T(n,k) read by rows; related to number of preorders.

A178514 Triangle read by rows: T(n,k) is the number of derangements of {1,2,...,n} having genus k (see first comment for definition of genus).

A211871 Number T(n,k) of permutations of n elements with no fixed points and largest cycle of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A219836 Triangular array counting derangements by number of descents.

A259784 Number T(n,k) of permutations p of [n] with no fixed points where the maximal displacement of an element equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A271697 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j-1,-n-1)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.

### A001710 Order of alternating group A_n, or number of even permutations of n letters.

A008970 Triangle T(n,k) = P(n,k)/2, n >= 2, 1<=k<n, of one-half of number of permutations of 1..n such that the differences have k runs with the same signs.

A010028 Triangle read by rows: T(n,k) = one-half the number of permutations of length n with exactly n-k rising or falling successions, for n >= 1, 1 <= k <= n. T(1,1) = 1 by convention.

A049444 Generalized Stirling number triangle of first kind.

A065826 Triangle with T(n,k)=k*E(n,k) where E(n,k) are Eulerian numbers A008292.

A086856 Triangle read by rows: T(n,k) = one-half number of permutations of length n with exactly k rising or falling successions, for n >= 1, 0 <= k <= n-1. T(1,0) = 1 by convention.

A122843 Triangle read by rows: T(n,k) = the number of ascending runs of length k in the permutations of [n] for k <= n.

A128612 Triangle T(n,k) read by rows: number of permutations in [n] with exactly k ascents that have an even number of inversions.

A128613 Triangle T(n,k) read by rows: number of permutations in [n] with exactly k ascents that have an odd number of inversions.

A132178 Triangle read by rows: T(n,m) is the number of cyclic permutations of [n] in which m of successive numbers add to a prime. 0<=m<=n, read by rows n>=0.

A136124 Triangle read by rows: T(n,k)=(-1)^(n+k)*Sum(s(n,j),j=1..k), where s(n,j) are the signed Stirling numbers of the first kind (n>=2; 1<=k<=n-1; s(n,j)=A008275(n,j)).

A137339 A triangular sequence from a functional coefficient expansion of a raising factorial type: p(x,t)=1/(1-t)^(m*x);m=3.

A141689 Average of Eulerian numbers (A008292) and Pascal's triangle (A007318): t(n,m)=(A008292(n,m)+A007318(n,m))/2.

A143491 Unsigned 2-restricted Stirling numbers of the first kind.

A144696 Triangle of 2-restricted Eulerian numbers.

A145224 Triangle read by rows: T(n,k) is the number of even permutations (of an n-set) with exactly k fixed points.

A145225 T(n,k) is the number of odd permutations (of an n-set) with exactly k fixed points.

A145324 Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....

A159930 Triangle read by rows: a(1,1)=1. a(m,n) = a(m-1,n) + (sum of all terms in row m-1), for n<m. a(m,m) = sum of all terms in row m-1.

A162608 Triangle read by rows in which row n lists n+1 terms, starting with n!, such that the difference between successive terms is also equal to n!.

A168391 Worpitzky form polynomials for the Narayana triangle A001263(n,k):p(x,n) = Sum[A001263(n,k)*Binomial[x + k - 1, n - 1], {k, 1, n}]

A179457 Triangle read by rows: number of permutation trees of power n and width <= k.

A237996 Triangular array read by rows. T(n,k) is the number of even permutations of {1,2,...,n} that have exactly k cycles, n>=0,0<=k<=n.

A249796 Triangle T(n,k), n>=3, 3<=k<=n, read by rows. Number of ways to make n selections without replacement from a circular array of n unlabeled cells (ignoring rotations and reflection), such that the first selection of a cell adjacent to previously selected cells occurs on the k-th selection.

### A000085 Number of self-inverse permutations on n letters, also known as involutions; number of Young tableaux with n cells.

A047884 Triangle of numbers a(n,k) = number of Young tableaux with n cells and k rows (1<=k<=n); also number of self-inverse permutations on n letters in which the length of the longest scattered (i.e. not necessarily contiguous) increasing subsequence is k.

A049403 A triangle of numbers related to triangle A030528.

A066325 Coefficients of unitary Hermite polynomials He_n(x).

A073278 A triangle constructed from the coefficients of the n-th derivative of the normal probability distribution function.

A099174 Triangle read by rows: coefficients of modified Hermite polynomials.

A104556 Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.

A111924 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,1) for n >= 1.

A122848 Exponential Riordan array (1,x(1+x/2)).

A144299 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,0) for n >= 0.

A152736 Triangle read by rows: M*Q, where M = an infinite lower triangular matrix with A140456 in every column: (1, 1, 1, 3, 7, 23, 71,...) and Q = a matrix with A000085 as the main diagonal the rest zeros.

A157391 A partition product of Stirling_1 type [parameter k = 1] with biggest-part statistic (triangle read by rows).

A161126 Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} having k descents (n>=1; 0<=k<n).

A178249 Table T(n,k) counts the involutions of n with longest increasing contiguous subsequence of length k.

A178515 Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} having genus k (see first comment for definition of genus).

A230698 Triangle read by rows: T(n,k) = T(n-1,k-1) + n*T(n-2,k); T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0.

A238121 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having exactly k descents, n>=0, 0<=k<=n.

A238123 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having k largest parts, n>=0, 0<=k<=n.

A238125 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having exactly k flat steps, n>=0, 0<=k<=n.

A238128 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having largest descent k, n>=0, 0<=k<=n.

A238129 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having largest ascent k, n>=0, 0<=k<=n.

A238707 Number T(n,k) of ballot sequences of length n having difference k between the multiplicities of the smallest and the largest value; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A238727 Number T(n,k) of standard Young tableaux with n cells where k is the largest value in the last row; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A238750 Number T(n,k) of standard Young tableaux with n cells and largest value n in row k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A238794 Number T(n,k) of standard Young tableaux with n cells and k as last value in the first row; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A238802 Number T(n,k) of standard Young tableaux with n cells where k is the length of the maximal consecutive sequence 1,2,...,k in the first column; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A238889 Number T(n,k) of self-inverse permutations p on [n] where the maximal displacement of an element equals k: k = max_{i=1..n} |p(i)-i|; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A239145 Number T(n,k) of self-inverse permutations p on [n] where the minimal transposition distance equals k (k=0 for the identity permutation); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

### A033312 a(n) = n! - 1.

A008291 Triangle of rencontres numbers.

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

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

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

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.

A125714 Alfred Moessner's factorial triangle.

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).

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)].

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.

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).

A211370 Array read by antidiagonals: T(m,n) = Sum( n <= i <= m+n-1 ) i!.

### A038507 n! + 1.

A146540 The PolyLog functional part of A008292 (the Eulerian numbers) is treated as a curvature to give a set of polynomial triangle sequence coefficients: p(x,n)=Sum[A008292(n,m)*x^(m-1),{m,0,n}]; q(x,n)=k from Solve[FullSimplify[ExpandAll[p[x, n]/(x - 1)^n]] - (1 + k/x^2) == 0, k].

### A005095 a(n) = n! + n.

A135723 A122890 + A000012 - I, I = Identity matrix.

### A005096 n! - n.

A176487 Triangle t(n,m) = binomial(n,m) + A008292(n+1,m+1)-1 read by rows.

A231210 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of some of the consecutive patterns 123, 1432, 2431, 3421; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

### A235378 a(n) = (-1)^n*(n!-(-1)^n), a sequence linking rencontres numbers r(n) with sum(k>=1, 1/((k+n)*k!) ) = (a(n)+(-1)^(n+1)*e*r(n))/n.

A132795 Triangle of Gely numbers, read by rows.

## Constant or linear factor

### A052849 a(0) = 0; a(n+1) = 2*n! (n >= 0).

A008518 Triangle of Eulerian numbers with rows multiplied by 1+x.

A128564 Triangle, read by rows, where T(n,k) equals the number of permutations of {1..n+1} with [(nk+k)/2] inversions for n>=k>=0.

A137513 Triangle read by rows: the coefficients of the Mittag-Leffler polynomials.

A141903 A linear combination of A008292 and A130595: t(n,m)=2*A008292(n,m)- A130595(n,m).

A142156 Triangle T(n,k)= n! if k=0, T(n,k) = -(n-k)!*A003319(k) if k>0.

A158471 Stirling-like triangle by rows generated from (x-1)*(x-1)*(x-2)*(x-3)*(x-4)*...

A198895 Triangle of coefficients arising in expansion of n-th derivative of tan(x) + sec(x).

A227342 Expansion of (1 - t)*(1 + t)^x.

### A001563 a(n) = n*n! = (n+1)! - n!.

A094485 T(n,k) = Stirling1(n+1,k+1)-Stirling1(n,k), n>=1, 0<=k<=n-1.

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).

A122844 Triangle read by rows: T[n,k] = the number of ascending runs of length at least k in the permutations of [n] for k <= n.

A130493 Triangle read by rows in which row n contains n! repeated n times.

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.

A185105 Number of entries in the k-th cycles of all permutations of {1,2,..,n}; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

## Double growth (2n)! and variants

### A010050 (2n)!.

A160486 Triangle of polynomial coefficients related to the o.g.f.s. of the RBS1 polynomials

A193639 Triangle T(n,k) of ways n couples can sit in a row with exactly k of them together

### A009445 a(n) = (2*n+1)!.

A214299 Triangle d_k(n) read by rows: number of n-th order Feynman diagrams with k interactions, 0<=k<=n.

### A001147 Double factorial of odd numbers: a(n) = (2*n-1)!! = 1*3*5*...*(2*n-1).

A008517 Second-order Eulerian triangle T(n,k), 1<=k<=n.

A039683 Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).

A055140 Triangle: matchings of 2n people with partners (of either sex) such that exactly k couples are left together.

A059364 Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.

A079267 d(n,s) = number of perfect matchings on {1, 2, ..., n} with k short pairs.

A088996 Triangle T(n,k) read by rows, given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938.

A099755 Triangle read by rows: T(n,0)=1, T(n,n)=(2*n-1)!!+1, T(m,n)=2*(m-n)*T(m-1,n-1)+(2*n+2)*T(m-1,n).

A102625 Triangle read by rows: T(n,k) is the sum of the weights of all vertices labeled k at depth n in the Catalan tree (1<=k<=n+1, n>=0).

A112007 Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.

A122774 Triangle of bifactorial numbers, n B m = (2(n-m)-1)!! (2(n-1))!! / (2(n-m))!!, read by rows.

A127160 Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,2,3,4,5,6,...] DELTA [1,1,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938.

A142070 A triangle of coefficients of rational root polynomials: p(x,n)=Product[(i + 1)*x - i, {i, 1, n}].

A156919 Table of coefficients of polynomials related to the Dirichlet eta function.

A161119 Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k cycles with entries of opposite parities (0<=k<=n).

A161198 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2)

A163936 Triangle related to the o.g.f.s. of the right hand columns of A130534 (E(x,m=1,n))

A163937 Triangle related to the o.g.f.s. of the right hand columns of A028421 (E(x,m=2,n))

A185410 A decomposition of the double factorials A001147.

A185411 A triangular decomposition of the double factorial numbers A001147.

A193561 Augmentation of the triangle A004736. See Comments.

A201637 Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.

A230696 Triangle read by rows related to double factorial of odd numbers (A001147).

### A000165 Double factorial of even numbers: (2n)!! = 2^n*n!.

A021012 Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).

A028338 Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x+2*n-1).

A039757 Triangle of coefficients in expansion of (x-1)(x-3)(x-5)...(x-(2n-1)).

A039758 Triangle of B-analogs of Stirling numbers of first kind.

A059366 Triangle T(m,s), m >= 0, 0<=s<=m, arising in computation of certain integrals.

A060187 Triangle read by rows: Eulerian numbers of type B, T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n-2*k+1)*T(n-1, k-1) + (2*k-1)*T(n-1, k).

A076256 Coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the constant term.

A076257 Coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the coefficient of the highest power of x.

A109281 Triangle T(n,k) of elements of n-th Weyl group of type B whose reduced word uses n-k generators.

A109692 Triangle of coefficients in expansion of (1+x)(1+3x)(1+5x)(1+7x)...(1+(2n-1)x).

A138076 A signed version of A060187 obtained by taking the Z-transform of p(t,x)=Exp[t*(1+2*x)].

A146543 The LerchPhi functional part of A060187 MacMahon numbers is treated/ solved for as a curvature to give a set of polynomial triangle sequence coefficients: p(x,n)=Sum[A060187(n,m)*x^(m-1),{m,0,n}]; q(x,n)=k from Solve[FullSimplify[ExpandAll[p[x, n]/(x - 1)^n]] - (1 + k/x^2) == 0, k].

A152936 A vector recursion designed around a row sum of A000165: v(n)=if[odd,{1.n,n^2,...,2^n*n!-Sum2^m,{m,0,n/2-1}],2^n*n!-Sum2^m,{m,0,n/2-1}],...n^2.n,1},{1.n,n^2,...,2^n*n!-2Sum2^m,{m,0,n/2-1}],...n^2.n,1}].

A152969 Triangle read by rows: T(n,m)=floor[(m/n)*row(n)].

A171721 Coefficient expansion of: f(t,y)=((1 + y)/y - Exp[t/2])/(-1 + y*Exp[t])

A189507 Triangle read by rows: T(n,k) (n >= 0, 1 <= k <= n+1) are the signed Hultman numbers.

A193229 A double factorial triangle

A196347 Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).

A208057 Triangle by rows, generated from the odd integers and related to A000165.

### A002866 a(0) = 1; for n>0, a(n) = 2^(n-1)*n!.

A039762 Triangle of D-analogs of Stirling numbers of first kind.

A039763 Triangle of D-analogs of Stirling numbers of first kind.

A079638 Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.

A101845 Triangle formed by left half of A101842, read by rows.

A102012 Triangle formed by right half of A101842, read by rows.

A112226 Table T(n,k) of number of elements of Weyl group of type D of order 2^{n-1} n! such that a reduced word uses exactly n-k distinct simple reflections 0 <= k <= n, n>=1.

A131222 Exponential Riordan array [1, log((1-x)/(1-2x)].

A156992 Triangle T(n,k) = n!*binomial(n-1,k-1) read by rows, 1<=k<=n.

A257609 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.

A262226 Eulerian numbers of type D, the primary type.

A262227 Eulerian numbers of type D, the complementary type.

## Binomial factors, Catalan products, linked bipermutations

### n! * Catalan(n) = A001761 = (2n)!/(n+1)!

A050145 T(n,k)=M(2n,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.

A050155 Triangle T(n,k), k>=0 and n>=1, read by rows defined by: T(n,k) = (2k+3)*binomial(2n,n-k-1)/(n+k+2).

A062196 Coefficient triangle of certain polynomials N(2; m,x).

A128821 Triangle T(n,k), 1<=k<=n, read by rows defined by :T(n,k)=C(n,k)*C(n-1,k-1)+C(n,k-1)*C(n-1,k)where C(n,k)=A007318(n,k) .

A132812 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).

A136534 A001263 * A128064 (unsigned).

A136536 A001263 * A128064 * A000012 as infinite lower triangular matrices.

A141811 Partial Catalan numbers: triangle read by rows n = 1, 2, 3, ... and columns k = 0, 1, ..., n-1.

A172417 n*Catalan number(n+1) triangle.

A176992 Triangle T(n,m) = binomial(2n-k+1,n+1) read by rows, 0<=k<=n.

A178300 Triangle T(n,k) = binomial(n+k-1,n) read by rows, 1<=k<=n.

### (n+1)! * Catalan(n) = A001813 = (2n)!/(n)!

A038455 A Jabotinsky-triangle related to A006963.

A064307 Triangle of coefficients of certain numerator polynomials N(n,x).

A156653 Coefficients of a higher level infinite sum polynomial: p(x,n)=(1 - x)^(2n + 1)/((n + 1)*x^n)*Sum[(k + 1)^n*Binomial[k, n]*x^ k, {k, 0, Infinity}].

A220883 Triangle read by rows: row n gives coefficients of expansion of Product_{k = 1..n-1} ((n + 1)*x + k), starting with lowest power.

A260687 Triangular array with n-th row giving coefficients of polynomial Product_{k = 2..n} (k + n*t) for n >= 1.

## Square growth, bipermutations

### A001044 a(n) = (n!)^2.

A027477 Square of lower triangular normalized first kind Stirling matrix.

A027537 Square of the lower triangular normalized Eulerian number matrix.

A192721 The number of pairs of permutations in the product group S_n X S_n with k common descents, n >= 1 and 0 <= k <= n-1.

A226780 Triangular array read by rows. T(n,k) is the number of 2 tuple lists of length n that have exactly k coincidences; n >= 0, 0 <= k <= n.

### A010790 a(n) = n!*(n+1)!.

A129274 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk+k) in the squared q-factorial of n+1.

A217940 Triangle read by rows: coefficients of polynomials Q_n(x) arising in study of Riemann zeta function.

## Others

### A032107 Number of reversible strings with n labeled beads of 2 colors.

A066094 Type D Eulerian triangle.

### A052593 E.g.f. 1/(1-x-x^4).

A145142 Triangle T(n,k), n>=1, 0<=k<=n-1, read by rows: T(n,k)/(n-1)! is the coefficient of x^k in polynomial p_n for the n-th row sequence of A145153.

### A188588 Row sums of 1-Euler triangle A188587.

A188587 1-Euler triangle.