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Triangles Theme Endofunctions
This is the latest A000312 Number of labeled mappings from n points to themselves (endofunctions): n^n.
A019575 Place n distinguishable balls in n boxes (in n^n ways); let T(n,k) = number of ways that the maximum in any box is k, for 1<=k<=n; sequence gives triangle of numbers T(n,k).
A038675 Triangle read by rows: T(n,k)=A(n,k)*binomial(n+k-1,n), where A(n,k) are the Eulerian numbers (A008292).
A055134 Triangle read by rows: T(n,k) = number of labeled endofunctions on n points with k fixed points.
A060281 Triangle T(n,k) read by rows giving number of labeled mappings (or functional digraphs) from n points to themselves (endofunctions) with k cycles, k=1..n.
A066320 Triangle: T(n,k)=C(n,k)*k^k*(n-k)^(n-k-1) k=0..n-1.
A066324 Number of endofunctions on n labeled points constructed from k rooted trees.
A071208 Triangular array T(n,k) read by rows, giving number of labeled free trees such that the root is smaller than all its children, with respect to the number n of vertices and to the number k of decreasing edges.
A071210 Triangular array T(n,k) read by rows, giving number of labeled free trees such that the root is smaller than all its children, with respect to the number n of vertices and to the degree k of the root.
A071211 Triangular array T(n,k) read by rows, giving number of labeled free trees such that the root is smaller than all its children, with respect to the number n of vertices and to the label k of the root.
A075856 Triangle formed from coefficients of the polynomials p(1)=x, p(n+1)=(n+x*(n+1))*p(n)+x*x*diff(p(n),x).
A080524 Triangle read by rows in which the n-th row contains n distinct numbers whose sum is n^n. The numbers are terms of an arithmetic progression with a common difference 1 or 2 respectively accordingly as n is odd or even.
A090657 Triangle read by rows: T(n,k) = number of functions from [1,2,...,n] to [1,2,...,n] such that the image contains exactly k elements (0<=k<=n).
A101817 Triangle read by rows: T(n,h) = number of functions f:{1,2,...,n}->{1,2,...,n} such that |Image(f)|=h; h=1,2,...,n, n=1,2,3,... . Essentially A090657, but without zeros.
A101820 Triangle read by rows: T(n,h)/(n-1), where T is the array in A101819.
A111568 Triangle read by rows: row n contains n terms of the arithmetic progression having first term 1 and common difference 2[n^(n-1)-1]/(n-1).
A137370 Brahmagupta matrix in a Markov matrix recursion produces a set of polynomials: the special values of x->Sqrt[z];y->1;t->n gives a set of polynomials as determinants. The triangular sequence of the Coefficients of these polynomials are except for signs A055134.
A174552 Triangular array T(n,k): The differences in the columns of A174551.
A199656 Triangular array read by rows, T(n,k) is the number of functions from {1,2,...,n} into {1,2,...,n} with maximum value of k.
A206823 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} with exactly k elements x such that |f^(-1)(x)| = 1; n>=0, 0<=k<=n.
A209324 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} whose largest component has exactly k nodes; n>=1, 1<=k<=n.
A216242 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} with a height of k; n>=1, 0<=k<=n-1.
A216971 Triangle read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} that have exactly k nonrecurrent elements mapped to some (one or more) recurrent element. n >= 1, 0 <= k <= n-1.
A219694 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n} -> {1,2,...,n} that have exactly k nonrecurrent elements; n>=1, 0<=k<=n-1.
A219859 Triangular array read by rows: T(n,k) is the number of endofunctions, functions f:{1,2,...,n}->{1,2,...,n}, that have exactly k elements with no preimage; n>=0, 0<=k<=n.
A220234 Triangular array read by rows. T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} that have exactly k recurrent elements whose preimage contains only one element, n>=0, 0<=k<=n.
A225753 Triangle of transformations with k monotonic runs.
A228154 T(n,k) is the number of s in {1,...,n}^n having longest contiguous subsequence with the same value of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
A228617 T(n,k) is the number of s in {1,...,n}^n having shortest run with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
A231536 Triangular array read by rows. T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} whose functional digraph has exactly k nodes such that no nonrecurrent element is mapped into it. n>=1, 1<=k<=n.
A239098 Triangle read by rows: T(0,0)=1; T(m,0)=0; otherwise T(m,n) = (m-1)*T(m-1,n)+(m-1+n)*T(m-1,n-1).
A241981 Number T(n,k) of endofunctions on [n] where the largest cycle length equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
A243203 Terms of a particular integer decomposition of N^N.
A244121 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).
A244137 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).
A245667 Number T(n,k) of sequences in {1,...,n}^n with longest increasing subsequence of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
A245687 Number T(n,k) of endofunctions on [n] such that the minimal cardinality of the nonempty preimages equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
A245692 Number T(n,k) of endofunctions f 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.
A245733 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality k exists and, if j is the largest value with a nonempty preimage, the preimage cardinality of i is >=k for all i<=j and equal to k for at least one i<=j; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
A246049 Number T(n,k) of endofunctions on [n] where the smallest cycle length equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
A264902 Number T(n,k) of defective parking functions of length n and defect k; triangle T(n,k), n>=0, 0<=k<=max(0,n-1), read by rows.
A000169 Number of labeled rooted trees with n nodes: n^(n-1).
A019576 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives triangle of numbers f(n,k)/n.
A034855 Triangle read by rows giving number of rooted labeled trees with n >= 2 nodes and height d >= 1.
A054589 Table related to labeled rooted trees, cycles and binary trees.
A055302 Triangle of labeled rooted trees with n nodes and k leaves, n>=1, 1<=k<=n.
A067948 Triangle of labeled rooted trees according to the number of increasing edges.
A071207 Triangular array T(n,k) read by rows, giving number of labeled free trees with n vertices and k children of the root that have a label smaller than the label of the root.
A101818 Triangle read by rows: (1/n)*T(n,h), where T(n,h) is the array in A101817.
A108267 Triangle, read by rows, where row n has g.f.: (1-x)^(n+1)*[Sum_{j=0..n} C(n+n*j+j,n*j+j)*x^j].
A122525 Triangle read by rows: G(s,rho) = ((s-1)!/s)*Sum(((s-i)/i!)*(s*rho)^i, i=0..(s-1)).
A174551 Triangular array T(n,k): functions f:{1,2,...,n}-> {1,2,...,n} such that each of k fixed (but arbitrary) elements are in the image of f.
A185390 Triangular array read by rows. T(n,k) is the number of partial functions on n labelled objects in which the domain of definition contains exactly k elements such that for all i in {1,2,3,...}, (f^i)(x) is defined.
A206429 Triangular array read by rows. T(n,k) is the number of rooted labeled trees on n nodes such that the root node has degree k. n>=2, 1<=k<=n-1.
A216520 Triangular array read by rows, T(n,k) = number of partial functions on {1,2,...,n} with exactly k cycles.
A231602 Triangular array read by rows: T(n,k) is the number of rooted labeled trees on n nodes that have exactly k nodes with outdegree = 1, n>=1, 0<=k<=n-1.
A243594 Triangle read by rows: T(n,k) = coefficient of [x^(n-k)] in the expansion of the polynomial (x+n)^n.
A259334 Triangle read by rows: T(n,k) = k*(n-1)!*n^(n-k-1)/(n-k)!, 1 <= k <= n.
A007778 n^(n+1).
A210457 Triangular array read by rows: T(n,k) is the number of elements x in {1,2,...,n} such that |(f^-1)(x)| = k over all functions f:{1,2,...,n}->{1,2,...,n}; n>=0, 0<=k<=n.
