A224334 -id:A224334 - cp's OEIS frontend

A224114 T(n,k) = Number of idempotent n X n 0..k matrices of rank 2.

0, 0, 1, 0, 1, 21, 0, 1, 51, 282, 0, 1, 93, 1434, 3070, 0, 1, 147, 4182, 30410, 29505, 0, 1, 213, 9762, 136990, 546465, 260967, 0, 1, 291, 19110, 467290, 3784335, 8815317, 2175236, 0, 1, 381, 34602, 1240310, 18925485, 94529631

Offset: 1

R. H. Hardin, Apr 09 2013

Table starts:
.......0.......0........0........0.......0.....0...0.0.0
.......1.......1........1........1.......1.....1...1.1
......21......51.......93......147.....213...291.381
.....282....1434.....4182.....9762...19110.34602
....3070...30410...136990...467290.1240310
...29505..546465..3784335.18925485
..260967.8815317.94529631
.2175236

			Some solutions for n=3, k=4:
..1..0..2....0..1..3....0..0..1....0..0..0....0..0..0....1..0..0....1..3..0
..0..1..3....0..1..0....0..1..0....3..1..0....0..1..0....0..1..1....0..0..0
..0..0..0....0..0..1....0..0..1....3..0..1....4..0..1....0..0..0....0..3..1

Row 3 is A224334.

A276999 Triangle read by rows, T(n,k) = n^k - 2^(k/2)*KummerU(-k/2,1/2,n^2/2) for 0<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 9, 0, 0, 1, 12, 93, 0, 0, 1, 15, 147, 1175, 0, 0, 1, 18, 213, 2070, 17835, 0, 0, 1, 21, 291, 3325, 33825, 317667, 0, 0, 1, 24, 381, 5000, 58575, 635208, 6506647, 0, 0, 1, 27, 483, 7155, 94785, 1164429, 13536453, 150776397

Offset: 0

Peter Luschny, Oct 06 2016

East and Gray (p. 24) give a combinatorial interpretation of the numbers: A function f: Y -> X with Y <= X (<= inclusion) has a 2-cycle if there exists x, y in Y with x != y, f(x) = y and f(y) = x. Then T(n,k) = card({f: [k] -> [n]: f has 2-cycles}).

			Triangle begins:
0;
0, 0;
0, 0, 1;
0, 0, 1,  9;
0, 0, 1, 12,  93;
0, 0, 1, 15, 147, 1175;
0, 0, 1, 18, 213, 2070, 17835;
0, 0, 1, 21, 291, 3325, 33825,  317667;
0, 0, 1, 24, 381, 5000, 58575,  635208,  6506647;
0, 0, 1, 27, 483, 7155, 94785, 1164429, 13536453, 150776397;
.
For instance T(3,3) = 9 because there are 27 functions [3]->[3], 18 of which have
no 2-cycles. The 9 functions which have 2-cycles are (represented as [f(1), f(2),
f(3)]): [1, 3, 2], [2, 1, 1], [2, 1, 2], [2, 1, 3], [2, 3, 2], [3, 1, 1],
[3, 2, 1], [3, 3, 1], [3, 3, 2].

J. East, R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.

T(n,k) = n^k - A244490(n,k), T(n,3) = A008585(n) for n>=3, T(n,4) = A224334(n-1) for n>=4, T(n,5) = A127694(n-3) for n>=5.

T := (n,k) -> n^k - 2^(k/2)*KummerU(-k/2, 1/2, n^2/2):
seq(seq(simplify(T(n,k)), k=0..n), n=0..9);

Table[Simplify[n^k - 2^(-k/2) HermiteH[k, n/Sqrt[2]]], {n, 0, 10}, {k, 0, n}] // Flatten

def T(n, k):
    @cached_function
    def h(n, x):
        if n == 0: return 1
        if n == 1: return 2*x
        return 2*(x*h(n-1,x)-(n-1)*h(n-2,x))
    return n^k - h(k, n/sqrt(2))/2^(k/2)
for n in range(10):
    print([T(n,k) for k in (0..n)])

T(n,k) = n^k - 2^(-k/2)*HermiteH(k, n/sqrt(2)).
T(n,k) = n^k - Sum_{i=0..k/2} k!/((-2)^i*i!*(k-2*i)!)*n^(k-2*i).
T(n,k) = n^k*(1-hypergeom([-k/2, (1-k)/2], [], -2/n^2)) for n>=1.
T(n,k) ~ n^k*(((k-1)*k)/(2*n^2)-(k*(k^3-6*k^2+11*k-6))/(8*n^4)+(k*(k^5-15*k^4 +85*k^3-225*k^2+274*k-120))/(48*n^6)+O((1/n)^7)).

cp's OEIS Frontend

A224114 T(n,k) = Number of idempotent n X n 0..k matrices of rank 2.

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A276999 Triangle read by rows, T(n,k) = n^k - 2^(k/2)*KummerU(-k/2,1/2,n^2/2) for 0<=k<=n.

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