A214663 -id:A214663 - cp's OEIS frontend

A276837 Number A(n,k) of permutations of [n] such that for each cycle c the smallest integer interval containing all elements of c has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 6, 5, 1, 0, 1, 1, 2, 6, 12, 8, 1, 0, 1, 1, 2, 6, 24, 25, 13, 1, 0, 1, 1, 2, 6, 24, 60, 57, 21, 1, 0, 1, 1, 2, 6, 24, 120, 150, 124, 34, 1, 0, 1, 1, 2, 6, 24, 120, 360, 399, 268, 55, 1, 0

Offset: 0

Alois P. Heinz, Sep 20 2016

The sequence of column k satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0.

			Square array A(n,k) begins:
  1, 1,  1,   1,    1,    1,    1,     1,     1, ...
  0, 1,  1,   1,    1,    1,    1,     1,     1, ...
  0, 1,  2,   2,    2,    2,    2,     2,     2, ...
  0, 1,  3,   6,    6,    6,    6,     6,     6, ...
  0, 1,  5,  12,   24,   24,   24,    24,    24, ...
  0, 1,  8,  25,   60,  120,  120,   120,   120, ...
  0, 1, 13,  57,  150,  360,  720,   720,   720, ...
  0, 1, 21, 124,  399, 1050, 2520,  5040,  5040, ...
  0, 1, 34, 268, 1145, 3192, 8400, 20160, 40320, ...

Alois P. Heinz, Antidiagonals n = 0..30, flattened
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019

Columns k=0-10 give: A000007, A000012, A000045(n+1), A214663, A276838, A276839, A276840, A276841, A276842, A276843, A276844.
Main diagonal gives A000142.
Cf. A263757, A276719.

A(n,k+1) - A(n,k) = A263757(n,k) for n>0.

A263597 T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and the median of every three consecutive elements nondecreasing.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 6, 5, 1, 2, 6, 12, 8, 1, 2, 6, 16, 25, 13, 1, 2, 6, 16, 41, 57, 21, 1, 2, 6, 16, 52, 108, 124, 34, 1, 2, 6, 16, 52, 164, 280, 268, 55, 1, 2, 6, 16, 52, 208, 476, 729, 588, 89, 1, 2, 6, 16, 52, 208, 676, 1428, 1908, 1285, 144, 1, 2, 6, 16, 52, 208, 800, 2208, 4308

Offset: 1

R. H. Hardin, Oct 22 2015

Table starts
..1....1....1.....1.....1.....1.....1.....1.....1.....1.....1.....1.....1.....1
..2....2....2.....2.....2.....2.....2.....2.....2.....2.....2.....2.....2.....2
..3....6....6.....6.....6.....6.....6.....6.....6.....6.....6.....6.....6.....6
..5...12...16....16....16....16....16....16....16....16....16....16....16....16
..8...25...41....52....52....52....52....52....52....52....52....52....52....52
.13...57..108...164...208...208...208...208...208...208...208...208...208...208
.21..124..280...476...676...800...800...800...800...800...800...800...800...800
.34..268..729..1428..2208..2900..3360..3360..3360..3360..3360..3360..3360..3360
.55..588.1908..4308..7696.10960.14024.16224.16224.16224.16224.16224.16224.16224
.89.1285.4969.12816.25508.40792.55492.69212.78088.78088.78088.78088.78088.78088

			Some solutions for n=6 k=4
..1....1....1....0....2....0....0....0....1....4....3....2....1....4....0....2
..2....2....0....1....1....5....3....1....5....1....1....0....4....0....1....0
..3....3....2....3....5....2....2....2....0....0....0....1....0....1....3....1
..0....0....4....4....3....3....1....5....2....2....2....5....3....3....5....3
..4....5....3....2....0....4....4....3....4....3....4....3....5....2....2....4
..5....4....5....5....4....1....5....4....3....5....5....4....2....5....4....5

R. H. Hardin, Table of n, a(n) for n = 1..516

Column 1 is A000045(n+1).

Column 2 is A214663.

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = a(n-1) +a(n-2) +3*a(n-3) +a(n-4)
k=3: a(n) = a(n-1) +a(n-2) +7*a(n-3) +2*a(n-4) +4*a(n-5) -a(n-7) -a(n-8)
k=4: [order 15]
k=5: [order 31]
k=6: [order 67]

A232164 Number of Weyl group elements, not containing an s_r factor, which contribute nonzero terms to Kostant's weight multiplicity formula when computing the multiplicity of the zero-weight in the adjoint representation for the Lie algebra of type C and rank n.

Original entry on oeis.org

0, 1, 1, 2, 6, 12, 25, 57, 124, 268, 588, 1285, 2801, 6118, 13362, 29168, 63685, 139057, 303608, 662888, 1447352, 3160121, 6899745, 15064810, 32892270, 71816436, 156802881, 342360937, 747505396, 1632091412, 3563482500, 7780451037, 16987713169, 37090703118

Offset: 0

Pamela E Harris, Nov 19 2013

Apart from the offset the same as A214663. - R. J. Mathar, Nov 27 2013

Apart from the initial 0, number of permutations of length n>=0 avoiding the partially ordered pattern (POP) {1>4} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the last element. - Sergey Kitaev, Dec 08 2020

			For n=4, a(4)= A232164(3) + A232164(2) + 3*A232164(1) + A232164(0) = 2+1+3*1+0=6.

P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
P. E. Harris, E. Insko, and L. K. Williams, The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula, arXiv preprint arXiv:1401.0055, 2013
B. Kostant, A Formula for the Multiplicity of a Weight, Proc Natl Acad Sci U S A. 1958 June; 44(6): 588-589.
László Németh and Dragan Stevanović, Graph solution of system of recurrence equations, Research Gate, 2023. See Table 1 at p. 3.
Kai Ting Keshia Yap, David Wehlau, and Imed Zaguia, Permutations Avoiding Certain Partially-ordered Patterns, arXiv:2101.12061 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).

a:=proc(n::nonnegint)
if n=0 then return 0:
elif n=1 then return 1:
elif n=2 then return 1:
elif n=3 then return 2:
else return
a(n-1)+a(n-2)+3*a(n-3)+a(n-4):
end if;
end proc:

CoefficientList[Series[x/(1 - x - x^2 -3 x^3- x^4),{x, 0, 30}], x] (* Vincenzo Librandi, Dec 31 2013 *)

Vec(-x/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Dec 31 2013

a(n) = A232164(n-1) + A232164(n-2) + 3*A232164(n-3) + A232164(n-4).
From Colin Barker, Dec 31 2013: (Start)
a(n) = a(n-1) + a(n-2) + 3*a(n-3) + a(n-4).
G.f.: -x/(x^4 + 3*x^3 + x^2 + x - 1). (End)

cp's OEIS Frontend

A276837 Number A(n,k) of permutations of [n] such that for each cycle c the smallest integer interval containing all elements of c has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A263597 T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and the median of every three consecutive elements nondecreasing.

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A232164 Number of Weyl group elements, not containing an s_r factor, which contribute nonzero terms to Kostant's weight multiplicity formula when computing the multiplicity of the zero-weight in the adjoint representation for the Lie algebra of type C and rank n.

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