A132647 -id:A132647 - cp's OEIS frontend

A370383 Number of permutations of [n] having no substring [k,k+1,k+2,k+3,k+4].

1, 1, 2, 6, 24, 119, 717, 5026, 40242, 362376, 3625081, 39885851, 478714416, 6224078292, 87145277160, 1307271652917, 20917481850667, 355612235468396, 6401234296266540, 121626707638142280, 2432586885636105251, 51085230669413519349, 1123891538655073251190

Offset: 0

Seiichi Manyama, Feb 17 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
D. M. Jackson and R. C. Read, A note on permutations without runs of given length, Aequationes Math. 17 (1978), no. 2-3, 336-343.

Cf. A000255, A002628, A132647, A184182, A370384.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*((x-x^5)/(1-x^5))^k))

G.f.: Sum_{k>=0} k! * ( (x-x^5)/(1-x^5) )^k.

a(n) = Sum_{k=0..4} A184182(n,k). - Alois P. Heinz, Feb 17 2024

A370384 Number of permutations of [n] having no substring [k,k+1,k+2,k+3,k+4,k+5].

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 719, 5037, 40306, 362802, 3628296, 39913080, 478970641, 6226733531, 87175347936, 1307641346772, 20922387099240, 355682119243320, 6402298503373917, 121643960874649867, 2432883613692550316, 51090627024035616300, 1123995015882951892680

Offset: 0

Seiichi Manyama, Feb 17 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
D. M. Jackson and R. C. Read, A note on permutations without runs of given length, Aequationes Math. 17 (1978), no. 2-3, 336-343.

Cf. A000255, A002628, A132647, A184182, A370383.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*((x-x^6)/(1-x^6))^k))

G.f.: Sum_{k>=0} k! * ( (x-x^6)/(1-x^6) )^k.

a(n) = Sum_{k=0..5} A184182(n,k). - Alois P. Heinz, Feb 17 2024

A370392 Number of permutations of [n] whose longest block is of length 3. A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions.

Original entry on oeis.org

0, 0, 0, 1, 2, 11, 63, 415, 3121, 26402, 248429, 2575936, 29198926, 359351878, 4773277246, 68078349863, 1037820312090, 16842621113247, 289946286959875, 5277826030457339, 101291053229162471, 2044252472193005928, 43283094591188747415, 959369370636209414390

Offset: 0

Seiichi Manyama, Feb 17 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Column k=3 of A184182.

Cf. A002628, A132647.

my(N=30, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=0, N, k!*x^k*(((1-x^3)/(1-x^4))^k-((1-x^2)/(1-x^3))^k))))

a(n) = A132647(n) - A002628(n).

G.f.: Sum_{k>=0} k! * x^k * ( ((1-x^3)/(1-x^4))^k - ((1-x^2)/(1-x^3))^k ).

cp's OEIS Frontend

A370383 Number of permutations of [n] having no substring [k,k+1,k+2,k+3,k+4].

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A370384 Number of permutations of [n] having no substring [k,k+1,k+2,k+3,k+4,k+5].

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Author

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Formula

A370392 Number of permutations of [n] whose longest block is of length 3. A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions.

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Author

Keywords

Links

Crossrefs

Programs

PARI

Formula