A184182 -id:A184182 - cp's OEIS frontend

A370390 Number of permutations of [n] whose longest block is of length 2. A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions.

0, 0, 1, 2, 10, 53, 334, 2428, 20009, 184440, 1881050, 21034905, 255967940, 3367720736, 47641219569, 721160081974, 11631770791362, 199159952915293, 3607908007376418, 68946510671942892, 1386140583681969289, 29247292475233307612, 646231776371742321826

Offset: 0

Seiichi Manyama, Feb 17 2024

nonn

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

Column k=2 of A184182.

Cf. A000255, A002628.

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

a(n) = A002628(n) - A000255(n-1).

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

A184180 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} whose shortest block is of length k (1 <= k <= n). A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 4512367 has 3 blocks: 45, 123, and 67. Its shortest block has length 2.

Original entry on oeis.org

1, 1, 1, 5, 0, 1, 22, 1, 0, 1, 117, 2, 0, 0, 1, 713, 5, 1, 0, 0, 1, 5026, 11, 2, 0, 0, 0, 1, 40285, 31, 2, 1, 0, 0, 0, 1, 362799, 73, 5, 2, 0, 0, 0, 0, 1, 3628584, 201, 11, 2, 1, 0, 0, 0, 0, 1, 39916243, 532, 20, 2, 2, 0, 0, 0, 0, 0, 1, 479000017, 1534, 40, 5, 2, 1, 0, 0, 0, 0, 0, 1, 6227016356, 4346, 82, 11, 2, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Emeric Deutsch, Feb 13 2011

Sum of entries in row n is n!.

T(n,1) = A184181(n).

			T(5,2) = 2 because we have 45123 and 34512.
Triangle starts:
    1;
    1, 1;
    5, 0, 1;
   22, 1, 0, 1;
  117, 2, 0, 0, 1;
  713, 5, 1, 0, 0, 1;
  ...

Cf. A000142, A000166, A184181, A184182.

d[0] := 1: for n to 40 do d[n] := n*d[n-1]+(-1)^n end do: T := proc (n, k) options operator, arrow: sum(binomial(n-(k-1)*m-1, m-1)*(d[m]+d[m-1]), m = 1 .. floor(n/k))-(sum(binomial(n-k*m-1, m-1)*(d[m]+d[m-1]), m = 1 .. floor(n/(k+1)))) end proc: for n to 13 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form

T[n_, k_] := With[{d = Subfactorial},
   Sum[Binomial[n-(k-1)*m-1, m-1]*(d[m] + d[m-1]), {m, 1, Floor[n/k]}] -
   Sum[Binomial[n-k*m-1, m-1]*(d[m] + d[m-1]), {m, 1, Floor[n/(k+1)]}]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 18 2024, after Maple code *)

T(n,k) = Sum_{m=1..floor(n/k)} binomial(n-(k-1)*m-1, m-1)*(d(m) + d(m-1)) - Sum_{m=1..floor(n/(k+1))} binomial(n-km-1, m-1)*(d(m) + d(m-1)), where d(j) = A000166(j) are the derangement numbers.

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

Original entry on oeis.org

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

A370390 Number of permutations of [n] whose longest block is of length 2. A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions.

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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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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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