A112839 -id:A112839 - cp's OEIS frontend

A112837 Large-number statistic from the enumeration of domino tilings of a 5-pillow of order n.

1, 1, 1, 1, 2, 3, 7, 12, 35, 87, 348, 1107, 5518, 22464, 150574, 817057, 7118856, 49644383, 560434040, 5142118400, 76370120248, 914476059335, 17638655014128, 274908897964359, 6936239946318204, 141510942505315328

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

nonn

A 5-pillow is a generalized Aztec pillow. The 5-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 5 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

			The number of domino tilings of the 5-pillow of order 6 is 1666=7^2*34. A112837(n)=7.

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

A112833 breaks down as A112834^2 times A112835, where A112835 is not necessarily squarefree.

3-pillows: A112833-A112835; 7-pillows: A112839-A112841; 9-pillows: A112842-A112844.

A114294 Modified Schroeder numbers for q=7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 5, 5, 5, 5, 2, 1, 13, 13, 13, 13, 5, 2, 1, 34, 34, 34, 34, 13, 5, 2, 1, 110, 110, 110, 110, 42, 16, 6, 2, 1, 393, 393, 393, 393, 150, 57, 21, 6, 2, 1, 1449, 1449, 1449, 1449, 553, 210, 77, 21, 6, 2, 1, 5390, 5390, 5390, 5390, 2057, 781

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Nov 21 2005

a(i,j) is the number of paths from (i,i) to (j,j) using steps of length (0,1), (1,0) and (1,1), not passing above the line y=x nor below the line y=3x/4. The Hamburger Theorem implies that we can use this table to calculate the number of domino tilings of an Aztec 7-pillow (A112839). To calculate this quantity, let P_n = the principal n X n submatrix of this array. If J_n = the back-diagonal matrix of order n, then A112839(n)=det(P_n+J_nP_n^(-1)J_n).

			The number of paths from (0,0) to (5,5) staying between the lines y=x and y=3x/4 using steps of length (0,1), (1,0) and (1,1) is a(0,5)=5.

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

See also A112833-A112844 and A114292-A114299.

cp's OEIS Frontend

A112837 Large-number statistic from the enumeration of domino tilings of a 5-pillow of order n.

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