cp's OEIS Frontend

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,0,2,0,3,0,4,0,5,0,6,0,7,0,...] DELTA [1,1,2,2,3,3,4,4,5,5,6,6,...] where DELTA is the operator defined in A084938; another version of A019538.

See also A019538: version with n > 0 and k > 0. - Philippe Deléham, Nov 03 2008

From Peter Bala, Jul 21 2014: (Start)

T(n,k) gives the number of (k-1)-dimensional faces in the interior of the first barycentric subdivision of the standard (n-1)-dimensional simplex. For example, the barycentric subdivision of the 1-simplex is o--o--o, with 1 interior vertex and 2 interior edges, giving T(2,1) = 1 and T(2,2) = 2.

This triangle is used when calculating the face vectors of the barycentric subdivision of a simplicial complex. Let S be an n-dimensional simplicial complex and write f_k for the number of k-dimensional faces of S, with the usual convention that f_(-1) = 1, so that F := (f_(-1), f_0, f_1,...,f_n) is the f-vector of S. If M(n) denotes the square matrix formed from the first n+1 rows and n+1 columns of the present triangle, then the vector F*M(n) is the f-vector of the first barycentric subdivision of the simplicial complex S (Brenti and Welker, Lemma 2.1). For example, the rows of Pascal's triangle A007318 (but with row and column indexing starting at -1) are the f-vectors for the standard n-simplexes. It follows that A007318*A131689, which equals A028246, is the array of f-vectors of the first barycentric subdivision of standard n-simplexes. (End)

This triangle T(n, k) appears in the o.g.f. G(n, x) = Sum_{m>=0} S(n, m)*x^m with S(n, m) = Sum_{j=0..m} j^n for n >= 1 as G(n, x) = Sum_{k=1..n} (x^k/(1 - x)^(k+2))*T(n, k). See also the Eulerian triangle A008292 with a Mar 31 2017 comment for a rewritten form. For the e.g.f. see A028246 with a Mar 13 2017 comment. - Wolfdieter Lang, Mar 31 2017

T(n,k) = the number of alignments of length k of n strings each of length 1. See Slowinski. An example is given below. Cf. A122193 (alignments of strings of length 2) and A299041 (alignments of strings of length 3). - Peter Bala, Feb 04 2018

The row polynomials R(n,x) are the Fubini polynomials. - Emanuele Munarini, Dec 05 2020

From Gus Wiseman, Feb 18 2022: (Start)

Also the number of patterns of length n with k distinct parts (or with maximum part k), where we define a pattern to be a finite sequence covering an initial interval of positive integers. For example, row n = 3 counts the following patterns:

(1,1,1) (1,2,2) (1,2,3)

(2,1,2) (1,3,2)

(2,2,1) (2,1,3)

(1,1,2) (2,3,1)

(1,2,1) (3,1,2)

(2,1,1) (3,2,1)

(End)

Regard A048994 as a lower-triangular matrix and divide each term A048994(n,k) by n!, then this is the matrix inverse. Because Sum_{k=0..n} (A048994(n,k) * x^n / n!) = A007318(x,n), Sum_{k=0..n} (A131689(n,k) * A007318(x,k)) = x^n. - Natalia L. Skirrow, Mar 23 2023

T(n,k) is the number of ordered partitions of [n] into k blocks. - Alois P. Heinz, Feb 21 2025

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is (strongly) alternating iff it is a weakly alternating anti-run.

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. A sequence is alternating iff it is a weakly alternating anti-run.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). An alternating pattern is necessarily an anti-run (A005649).

The version with twins (A344605) is identical to this sequence except with a(2) = 3 instead of 2.

From Gus Wiseman, Jan 16 2022: (Start)

Conjecture: Also the number of weakly up/down patterns of length n, where a sequence is weakly up/down if it is alternately weakly increasing and weakly decreasing, starting with an increase. For example, the a(0) = 1 through a(3) = 6 weakly up/down patterns are:

() (1) (1,1) (1,1,1)

(2,1) (1,1,2)

(2,1,1)

(2,1,2)

(2,1,3)

(3,1,2)

(End)

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is alternating in the sense of A025047 iff it is a weakly alternating anti-run.

A prime index of n is a number m such that prime(m) divides n. For n > 1, the multiset of prime factors of n is row n of A027746. The prime indices A112798 can also be used.

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

This sequence counts compositions that are weakly but not strongly alternating; also weakly alternating non-anti-run compositions.

First differs from 2 * A326291 at a(90) = 4, A326291(90) = 3.

The first odd term is a(144) = 7, whose non-weakly alternating permutations are shown in the example below.

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is alternating in the sense of A025047 iff it is a weakly alternating anti-run.

For n > 1, the multiset of prime factors of n is row n of A027746. The prime indices A112798 can also be used.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). An alternating pattern is necessarily an anti-run (A005649).

Conjecture: Also the number of non-weakly up/down (or down/up) patterns of length n. For example:

- The a(3) = 7 non-weakly up/down patterns:

(121), (122), (123), (132), (221), (231), (321)

- The a(3) = 7 non-weakly down/up patterns:

(112), (123), (211), (212), (213), (312), (321)

- The a(3) = 7 non-alternating patterns (see example for more):

(111), (112), (122), (123), (211), (221), (321)

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

This sequence ranks compositions that are weakly but not strongly alternating.