cp's OEIS Frontend

Number of ways to put numbers 1, 2, ..., n*(n+1)/2 in a triangular array of n rows in such a way that each row is increasing. Also number of ways to choose groups of 1, 2, 3, ..., n-1 and n objects out of n*(n+1)/2 objects. - Floor van Lamoen, Jul 16 2001

a(n) is the number of ways to linearly order the multiset {1,2,2,3,3,3,...n,n,...n}. - Geoffrey Critzer, Mar 08 2009

Also the number of distinct adjacency matrices in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017

From Tilman Piesk, Oct 28 2014: (Start)

Number of permutations of a multiset that contains m different elements n times. These multisets have the signatures A249543(m,n-1) for m>=1 and n>=2.

In an m-dimensional Pascal tensor (the generalization of a symmetric Pascal matrix) P(x1,...,xn) = (x1+...+xn)!/(x1!*...*xn!), so the main diagonal of an m-dimensional Pascal tensor is D(n) = (m*n)!/(n!^m). These diagonals are the rows of this array (with m>0), which begins like this:

m\n:0 1 2 3 4 5

0: 1 1 1 1 1 1 ... A000012;

1: 1 1 1 1 1 1 ... A000012;

2: 1 2 6 20 70 252 ... A000984;

3: 1 6 90 1680 34650 756756 ... A006480;

4: 1 24 2520 369600 63063000 11732745024 ... A008977;

5: 1 120 113400 168168000 305540235000 623360743125120 ... A008978;

6: 1 720 7484400 137225088000 3246670537110000 88832646059788350720 ... A008979;

with columns: A000142 (n=1), A000680 (n=2), A014606 (n=3), A014608 (n=4), A014609 (n=5).

A089759 is the transpose of this matrix. A034841 is its diagonal. A141906 is its lower triangle. A120666 is the upper triangle of this matrix with indices starting from 1. A248827 are the diagonal sums (or the row sums of the triangle).

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T(n,k) is the number of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1. - Alois P. Heinz, May 06 2013

Note that if n=p^k for p prime then a(n) is coprime to n (i.e., a(n) is not divisible by p).

a(n) is also the number of labelings for the simple graph K_n X K_n, the graph Cartesian product of the complete graph with itself. - Geoffrey Critzer, Oct 16 2016

a(n) is also the number of knockout tournament seedings with 2 rounds and n participants in each match. - Alexander Karpov, Dec 15 2017

The product of the terms of the n-th row is given by A034841.

Collection of partial binary matrices: 1 to n rows of length n and a total of n entries set to one in each partial matrix. - Olivier Gérard, Aug 08 2016

(n^2)!/(n!)^(n+1) is an integer for every n (see A057599). Hence k >= n+1. Conjecture: k=n+1 only when n is prime or a power of a prime.

Also number of linear unbranched n-4-catafusenes of C_{2v} symmetry.

Number of n-bead black-white reversible strings with k black beads; also binary grids; string is not palindromic. - Yosu Yurramendi, Aug 08 2008

The first seven columns are A004526, A002620, A006584, A032091, A032092, A032093, A032094. Row sums give essentially A032085. - Yosu Yurramendi, Aug 08 2008

From Álvar Ibeas, Jun 01 2020: (Start)

T(n, k) is the sum of odd-degree coefficients of the Gaussian polynomial [n, k]_q. The area below a NE lattice path between (0,0) and (k, n-k) is even for A034851(n, k) paths and odd for T(n, k) of them.

For a (non-reversible) string of k black and n-k white beads, consider the minimum number of bead transpositions needed to place the black ones to the left and the white ones to the right (in other words, the number of inversions of the permutation obtained by labeling the black beads by integers 1,...,k and the white ones by k+1,...,n, in the same order they take on the string). It is even for A034851(n, k) strings and odd for T(n, k) cases.

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